Section1.9Variables, Expressions, and Equations Chapter Review

Variables and Evaluating Expressions.

A variable represents an unknown quantity, or a quantity that can change. Algebra often uses \(x\) as the variable, but any letter or word can work as a variable. We also often use a letter that stands for something, like \(g\) for gas mileage.

When a variable represents a physical quantity, be clear about what are the units of measurement that apply. For example, \(g\) measuring gas mileage in miles per gallon is different from \(g\) measuring gas mileage in liters per kilometer.

An algebraic expression is any combination of variables and numbers using arithmetic operations like addition, multiplcation, etc. Algebraic expressions can be evaluated. This means substituting values in for the variable(s).

Be careful when evaluating an algebraic expression at a negative number. It often helps to wrap parentheses around the negative number you are substituting in.

Checkpoint1.9.1.

Identify a variable you might use to represent each quantity. Then identify what units would be most appropriate.

(a)

Let be the depth of a swimming pool, measured in .

Explanation.

The unknown quantity is depth, which starts with âdâ. We generally measure depth in feet for a swimming pool. (Meters is another reasonable unit.) So we could define this variable as:

âLet \(d\) be the depth of a swimming pool, measured in feet.â

(b)

Let be the weight of a dog, measured in .

Explanation.

The weight of the dog is the unknown quantity, and âweightâ starts with âwâ. We generally measure the weight of a dog in pounds. (Kilograms is another reasonable unit.) So we could define this variable as:

âLet \(w\) be be the weight of a dog, measured in pounds.â

Checkpoint1.9.2.

Evaluate the expression for the given value of the variable.

\({-8\mathopen{}\left(r+5\right)-2}\) for \(r=-3\)

Checkpoint1.9.3.

Evaluate the expression for the given value of the variable.

\({3m^{2}-8m-4}\) for \(m=-4\text{.}\)

Combining Like Terms.

In an algebraic expression, terms are pieces of the expression that are added together. For example, the terms in \(2x^2-5x+7\) are \(2x^2\text{,}\)\(-5x\text{,}\) and \(7\text{.}\)

Terms are different from factors, which are pieces of an algebraic expression that are multiplied together. For example, the factors of \(2x(x+5)\) are \(2\text{,}\)\(x\text{,}\) and \((x+5)\text{.}\)

Whenever terms are similar enough that they can be combined and simplified, they are called like terms. Like terms typicaly are some number multiplied by a variable, with the same variable in each term. But like terms can also have the same units of measure or the same radical factor in place of the variable. Or they can have the same power of a variable. Each of these expressions has two like terms:

Like terms arise in applications where it makes sense to add some things together, and it might happen that the terms you have to add are similar enough to be called like terms. For example, finding a perimeter of a polygon might be an application of combining like terms, if the sides of the polygon are each labeled as a number times some common variable.

Checkpoint1.9.4.

List the terms in each expression.

\({-0.1u-1.6d-d+6.2u}\)

Checkpoint1.9.5.

Simplify the expression by combining like terms if possible.

\({2b+Z+{\frac{8}{7}}Z}\)

Checkpoint1.9.6.

Write a simplified expression for the perimeter of the given shape (which is not drawn to scale).

Explanation.

The perimeter is the result from adding the six sides together: \(17f + 0.6J + J + f + f + 64J\text{.}\) There are two \(J\)-terms and two \(J\)-terms. We combine each pair of like terms and the perimeter is \({19f+65.6J}\text{.}\)

Comparison Symbols and Notation for Intervals.

The symbols used for comparing two quantities are as follows:

Symbol

Means

True

True

False

\(=\)

equals

\(13=13\)

\(\frac{5}{4}=1.25\)

\(5\reject{=}6\)

\(\gt\)

is greater than

\(13\gt11\)

\(\pi\gt3\)

\(9\reject{\gt}9\)

\(\geq\)

is greater than or equal to

\(13\geq11\)

\(3\geq3\)

\(10.2\reject{\geq}11.2\)

\(\lt\)

is less than

\(-3\lt8\)

\(\frac{1}{2}\lt\frac{2}{3}\)

\(2\reject{\lt}-2\)

\(\leq\)

is less than or equal to

\(-3\leq8\)

\(3\leq3\)

\(\frac{4}{5}\reject{\leq}\frac{3}{5}\)

\(\neq\)

is not equal to

\(10\neq20\)

\(\frac{1}{2}\neq1.2\)

\(\frac{3}{8}\reject{\neq}0.375\)

An interval is a collection of numbers on a number line that are all connected. We illustrate intervals with a number line, where some portion of the number line is shaded. To clear up whether or not an endpoint of the shaded region is part off the interval, we use brackets (to include that number) or parentheses (to exclude that number). An interval might extend forever in one direction, and then the graph uses an arrowhead. For example, here is a graph of the interval of all positive numbers.

And here is an interval with all numbers that are less than or equal to\(4\text{.}\)

There are two standard notations for how to communicate an interval of numbers. One is set-builder notation which is structured this way:

\begin{equation*}
\left\{\text{variable}\mid\text{condition variable must meet}\right\}
\end{equation*}

For example, \(\{x\mid x\gt0\}\) is set-builder notation for the interval of all positive numbers.

The other standard notation for an interval is interval notation. This notation identifies the left and right ends of an interval and just writes them down, separated by a comma. Brackets or parentheses indicate whether the end is included or not in the interval. For example, \((0,\infty)\) is the interval of all positive numbers, and \([0,\infty)\) is the interval of all non-negative numbers (meaning the positive numbers and also zero).

Checkpoint1.9.7.

Express the given interval in set-builder notation and interval notation.

Checkpoint1.9.8.

Express the given interval in set-builder notation and interval notation.

Checkpoint1.9.9.

Convert the given set-builder notation into a number line graph and interval notation.

\({\{ x \mid x \le -1 \}}\)

Checkpoint1.9.10.

Convert the given interval notation into a number line graph and set-builder notation.

\({\left(1,\infty \right)}\)

Equations, Inequalities, and Solutions.

An equation is a statement that two algebraic expressions are equal. There must be an equals sign (\(=\)) in between the two expressions. For example, \(x^2=x+2\) in an equation. An inequality is similar, but uses one of the five inequality symbols instead of an equals sign. An inequality is a statement about how the two expressions relate to each other.

When an equation or inequality only has one variable, a solution to the equation or inequality is a number that you can substitute in for the variable and it results in a true relation between pure numbers. For example, \(2\) is a solution to \(x^2=x+2\) because when you substitute \(2\) in for \(x\) and simplify each side, you have \(4\confirm{=}4\text{.}\) But for example, \(3\) is not a solution, since when you substitute \(3\) in for \(x\) and simplify each side, you have \(9\reject{=}5\text{.}\) The skill of checking whether or not a given number is a solution to an eqution or inequality is important.

A linear expression in one variable is an expression that simplifes to the form \(ax+b\) where \(a\) and \(b\) are specific numbers, but \(a\neq0\text{.}\) For example, \(\frac{2}{5}x+3\) and \(5(x+3)-2x\) are linear expressions.

A linear equation is a specific type of equation where the two sides of the equation are either both linear expressions, or one side is a linear expression and the other side is just a number. A linear inequality is similar but itâs an inequality, not an equation. Linear equations and inequalities are the focus of PartÂ I.

Checkpoint1.9.11.

Check if the given number is a solution to the given equation.

Is \({4}\) a solution to:

\(9x+{\frac{7}{5}}\)

\(=\)

\(8x+{\frac{49}{10}}\)

\(\wonder{=}\)

Checkpoint1.9.12.

Check if the given number is a solution to the given inequality.

Is \({5}\) a solution to:

\(6x-2\)

\(\lt\)

\(28\)

\(\wonder{\lt}\)

\(28\)

Checkpoint1.9.13.

Select the equations/inequalities that are linear with one variable.

\(\displaystyle 10y+4=34\)

\(\displaystyle q\sqrt{28}\leq54\)

\(\displaystyle 2Vy=-88\)

\(\displaystyle p^{2}+y^{2}\geq-31\)

\(\displaystyle \pi r^{2}\lt26\pi \)

\(\displaystyle \left|6.6z-9\right|=-25\)

None of the above

Solving One-Step Equations.

Suppose you would like to find the solution(s) to an equtaion like \(x+7=11\text{.}\) There is a formal process we can follow to do this. Since the variable has \(7\) added to it, we do the opposite action, subtracting \(7\text{,}\) so that the left side ends up being an isolated \(x\text{.}\) However, we want to end up with an equivalent equation to the one we started with. That means w want an equtaion with the same solution set. In order to do that we have to subtract \(7\) from both sides, not just the left side. After doing that we have \(x=4\) and it is straightforward to see that \(x\) must be \(4\text{.}\) The only solution is \(4\text{.}\)

Adding and subtracting are opposite operations. Multiplying and dividing are opposite operations. Keeping these pairings of opposites in mind, we can solve many small linear equations. According to FactÂ 1.5.12, we can always add or subtract any number on each side of an equation to obtain an equivalent equation. And we can always multiply or divide by any nonzero number to obtain an equivalent equation.

If a variable that you need to isolate is being multiplied by a fraction, then multiplying by the reciprocal of that fraction is one way to undo that. Of course, this is still an action that you must take to both sides of the equation.

The solution set to an equation is the collection of all numbers that are solutions. For the linear equation \(x+7=11\text{,}\) there was only one solution, so the solution set is a âcollectionâ that only has one number in it. Whenever a solution set only has a finite number of numbers in it, we use braces to write the solution set. In this case, the solution set is \(\{4\}\text{.}\) This is called set notation, not to be confused with set-builder notation.

The general process for solving equations is to:

Apply FactÂ 1.5.12 in a way that isolates the variable. This leads to a statment that the variable is some specific number.

Check that the number you found really works as a solution in the original equation. This will help you realize if you made a human arithmetic mistake somewhere in your process.

Summarize your findings. Once you have confirmed the solution, be explicit and write a statement of what the solution set is. Or if the algebra exercise had context, write something that communicates the contextual meaning of the solution.

Checkpoint1.9.14.

Solve the equation.

\({Z+19}={-5}\)

Checkpoint1.9.15.

Solve the equation.

\({{\frac{5}{12}}+f}={{\frac{7}{4}}}\)

Checkpoint1.9.16.

Solve the equation.

\({\frac{k}{5}}={-3}\)

Checkpoint1.9.17.

Solve the equation.

\({{\frac{9}{4}}q}={-{\frac{5}{6}}}\)

Checkpoint1.9.18.

In retail, an item has a wholesale price \(w\) that the store pays to obtain the item. The shelf price \(s\) is what a customer pays to buy the item. The âmarkup factorâ \(m\) is a number that explains what proportion of the shelf price is profit. For example if the markup factor is \(0.15\text{,}\) it means that \(15\%\) of the shelf price is profit for the store. These numbers are related by the formula \(s(1-m)=w\text{.}\)

Suppose the markup is \(45\%\) and the wholesale price is \({\$12.70}\text{.}\) Write an equation that could be used to find the shelf price. Then find the shelf price.

Solving One-Step Inequalities.

Solving linear inequalities is a lot like solving linear equations, but there are two important differences. One difference is that typically, the solution set is an interval of numbers. So it can be expressed using a number line graph, interval notation, or set-builder notation. But not using set notation as we do with solution sets to linear equations.

The other important difference is that whenever the solving process requires you to multiply or divide on each side by a negative number, the direction of the inequality symbol changes. For example when solving \(-2x\leq24\text{,}\) we would divide on each side by \(-2\text{.}\) And then we would have to change the direction of the inequality symbol and end with \(x\geq-12\text{.}\)

Checkpoint1.9.19.

Solve the inequality. Graph the solution set, and write the solution set using both interval notation and set-builder notation.

\({8B}\leq{-32}\)

Checkpoint1.9.20.

Solve the inequality. Graph the solution set, and write the solution set using both interval notation and set-builder notation.

\({H-15}\gt{-11}\)

Checkpoint1.9.21.

Solve the inequality. Graph the solution set, and write the solution set using both interval notation and set-builder notation.

\({-3M}\lt{15}\)

Checkpoint1.9.22.

Solve the inequality. Graph the solution set, and write the solution set using both interval notation and set-builder notation.

\({-{\frac{6}{5}}T}\geq{6}\)

Algebraic Properties and Simplifying Expressions.

The number \(0\) is called the additive identity because you can add \(0\) to any number and the value does not change. A numberâs additive inverse (or opposite) is the number you can add to it to get \(0\text{.}\) In other words, its negative. For example the additive inverse of \(8\) is \(-8\text{,}\) and the additiive inverse of \(-3.4\) is \(3.4\text{.}\)

The number \(1\) is called the multiplicative identity because you can multiply any number by \(1\) and the value does not change. A numberâs multiplicative inverse (or reciprocal) is the number you can multiply it by to get \(1\text{.}\) For example the multiplicative inverse of \(6\) is \(\frac{1}{6}\text{,}\) and the multiplicative inverse of \(-\frac{3}{2}\) is \(-\frac{2}{3}\text{.}\)

A commutative property allows you to write two numbers or expressions in the opposite order and have an equal result. For example, \(8+3 = 3+8\text{.}\) This illustrates that addition has the commutative property. And for example, \(4(9) = 9(4)\text{.}\) This illustrates that multiplication has the commutative property. Note that subtraction and diivision do not have the commutative property.

An associative property allows you to group three numbers or expressions in a different way without changing the order they are written. For example, \((8+3)+5 = 8+(3+5)\text{.}\) This illustrates that addition has the associative property. And for example, \((2\cdot3)(5) = 2(3\cdot5)\text{.}\) This illustrates that multiplication has the associative property. Note that subtraction and diivision do not have the associative property.

The distributive property of numbers combines multiplication/division with grouped addition/subtraction. For three numbers \(a\text{,}\)\(b\text{,}\) and \(c\) the following are all patterns that we call the distributive property:

(In the versions where there is division by \(a\text{,}\) we need \(a\neq0\text{.}\))

Technically, all these concepts above are the reasons why we can do things like combine like terms and simplify many kinds of algebra expressions. We learn about these concepts here, and yet you might find that you donât need to literally use them to succeed with solving algebra problems.

Checkpoint1.9.23.

Find the multiplicative inverse of \({-{\frac{2}{9}}}\)

Checkpoint1.9.24.

Find the additive inverse of \(-8\)

Checkpoint1.9.25.

Apply associativity to \({{\frac{1}{6}}+\left(k^{2}+k\right)}\text{.}\)

Checkpoint1.9.26.

Apply commutativity of multiplication to \({qL+l}\text{.}\)

Checkpoint1.9.27.

Apply the distributive property to \({5\mathopen{}\left(w-8\right)}\text{.}\)

Checkpoint1.9.28.

Simplify the given expression. Ideally, you are thinking about how the properties of algebra are helping you simplify.

\({9+2\mathopen{}\left(3B-4\right)}\)

Modeling with Equations and Inequalities.

When you have a âword problemâ in front of you, the first thing to do is read and re-read everything until you have an understanding of what the numbers really represent physically, and an understand of what exactly you are being asked to find. Once you have that understanding, clearly define a variable that represents whatever quantity you need to find. And clearly state what units of measure go with that variable, if there are any.

Many such application problems are ârate problemsâ. A rate is a measurement that tells us how much one quantity is changing with respect to how some other quantity is changing. They typically have fractional units, like ^{ft}â_{s}. The generic equation:

might be useful with a rate problem, to set up an equation where the solution to the equation answers the phsycial question you are trying tot answer.

Another application of algebra can be a âpercent problemâ where some quantity started out with some value, and then either iincreased or decreased by some percent and ended with a final value. If you are trying to solve for the initial value, this generic equation can help:

In this section, we are concerned with setting up these equations. Later we will actually solve them and answer the underlying applicatiion question. But it is challenging enough for now just to correctly set up these equations.

Occasionally, it is more appropriate to set up an inequality than an equation. Look for phrases like âis at mostâ, âneeds to be at leastâ, etc. And look for words that imply these meanings, like âmaximumâ, âminimumâ, etc. And use reading comprehension to understand when this is implied. For example if a person is working with a budget, they are required to spend no more than that amount. They could spend it all, or spend less.

Checkpoint1.9.29.

One of the tires on your car looks a little flat. You measure its air pressure and are alarmed to see it so low at \({24\ {\rm psi}}\text{.}\) You have a portable device that can pump air into the tire increasing the pressure at a rate of \({2.3\ {\textstyle\frac{\rm\mathstrut psi}{\rm\mathstrut min}}}\text{.}\) How long will it take to fill the tire to the manualâs recommended pressure of \({32\ {\rm psi}}\text{?}\) Write an equation to model this scenario. There is no need to solve it.

Checkpoint1.9.30.

A tool shed is for sale in a state where sales tax applies. The sales tax rate is \({6.8\%}\) and the total was \({\$368}\text{.}\) What was the price before sales tax? Set up an equation to answer this question. There is no need to solve it.

Checkpoint1.9.31.

An airplane was cruising at \({36000\ {\rm ft}}\text{,}\) and then began its descent. Because of some air traffic congestion, it will descend to \({9400\ {\rm ft}}\) and then cirlce the airport for a while. It descends at a rate of \({1000\ {\textstyle\frac{\rm\mathstrut ft}{\rm\mathstrut min}}}\text{.}\) A passenger notices the planee is still descending. Set up an inequality where thee solution set represents how much time might have passed since the plane started its descent. There is no need to solve it.

ExercisesReview Exercises for ChapterÂ 1

SectionÂ 1: Variables and Evaluating Expressions

1.

Identify a variable you might use to represent each quantity. Then identify what units would be most appropriate.

(a)

Let be the amount of time a person sleeps each night, measured in .

(b)

Let be the surface area of a patio, measured in .

Evaluating Expressions.

Evaluate the expression for the given value of the variable.

2.

\({c+4}\) for \(c=3\)

3.

\({f-9}\) for \(f=-6\)

4.

\({6h+9}\) for \(h=-3\)

5.

(a)

\({k^{3}}\) for \(k=4\)

(b)

\({k^{3}}\) for \(k=-5\)

6.

\(\displaystyle{\frac{8n+5}{2n}}\) for \(n=6\)

7.

\({{\frac{1}{6}}\mathopen{}\left(C+5\right)^{2}-2}\) for \(C=1\text{.}\)

8.

\({\left(4m\right)^{2}}\) for \(m=2\text{.}\)

9.

\({\left|p-8\right|+6}\) for \(p=7\text{.}\)

10.

\(\sqrt{\left(x_2-x_1\right)^2 + \left(y_2-y_1\right)^2}\) for \(x_1=0\text{,}\)\(x_2=5\text{,}\)\(y_1=6\text{,}\) and \(y_2=18\text{.}\)

11.

If you travel the road from Portland, OR to Boise, ID, and you have traveled \(x\) miles so far, you have \(431 - x\) miles left to go.

After traveling \(118\) miles, how far do you have left to go?

12.

If we want to represent a personâs target heart rate during exercise, weâd use the formula \(r=0.6(220-a)\) where \(a\) is the personâs age in years and \(r\) is their target heart rate in beats per minute (bpm).

What is the target heart rate of a person who is \(30\) years old?

13.

On Earth, if you throw a baseball straight up at speed \(v\) (in feet per second), the highest that it reaches is \(v^2/64+6\) feet above the ground.

If thrown straight up with speed \(59\) feet per second, how high will the baseball reach?

14.

The diagonal length \(D\) of a rectangle with side lengths \(L\) and \(W\) is given by \(D=\sqrt{L^2+W^2}\text{.}\)

Determine the diagonal length of a rectangle with side lengths \(16\) and \(12\text{.}\)

15.

There are \(3\) feet in a yard. Write an expression representing how many feet are in \(x\) yards.

16.

Suppose \(n\) family members live in a home, and some cousins, a family of five, comes to stay for a week. Give an expression for how many people live in that house during that week.

17.

An elementary school classroom needs a minimum of \(140\) square feet for the teacher plus a minimum of \(36\) square feet per student. Write an expression for the minimum square footage of a classroom with \(n\) students.

Simplify the expression by combining like terms if possible.

19.

\({Y - {\frac{10}{7}}Y}\)

20.

\({3.7c+0.1c+e}\)

21.

\({2j - {\frac{5}{4}}j+j}\)

22.

\({60s-3.5q+s+5.7q}\)

23.

Write a simplified expression for the perimeter of the given shape (which is not drawn to scale).

24.

Lucian and Esteban are co-owners of a pastry shop. Lucian bakes pies and Esteban bakes cakes. Lucian is able to bake \(p\) pies each day he works and Esteban is able to bake \(c\) cakes each day he works.

One month, Lucian worked \(16\) days and Esteban worked \(21\) days. That month they produced \({16p+21c}\) baked goods in total.

(a)

The next month, Lucian worked \(22\) days and Esteban worked \(20\) days. How many baked goods did they produce that month?

(b)

How many baked goods was that in total for those two months?

SectionÂ 3: Comparison Symbols and Notation for Intervals

25.

Decide if the comparison is true or false.

\({{\frac{10}{7}}} > {{\frac{3}{4}}}\)

Compare Two Numbers.

Decide if one given number is greater than, less than, or equal to another given number.

26.

\(-9.25\)\(-9.05\)

27.

\({1.225}\)\({{\frac{49}{40}}}\)

28.

Use the \(\gt\) symbol to arrange the following numbers in order from greatest to least. For example, your answer might look like \(4 \gt 3 \gt 2 \gt 1 \gt 0\text{.}\)

Express the given interval in set-builder notation and interval notation.

30.

Convert the given set-builder notation into a number line graph and interval notation.

\({\{ x \mid x \lt -5 \}}\)

31.

Convert the given interval notation into a number line graph and set-builder notation.

\({\left(-\infty ,-3\right)}\)

32.

In a battery, the negatively charged terminal is called the âanodeâ. Write an interval for the charge \(C\) that could be present on an anode.

33.

A water-based liquid has a âpHâ level. At room temperature, if the pH level is less than \(7\text{,}\) then the liquid is a âbaseâ. If it is greater than \(7\text{,}\) then the liquid is an âacidâ.

Write an interval for the pH level of a base.

SectionÂ 4: Equations, Inequalities, and Solutions

Check a Possible Solution to an Equation.

Check if the given number is a solution to the given equation.

34.

Is \({6}\) a solution to:

\(3x+8\)

\(=\)

\(17\)

\(\wonder{=}\)

\(17\)

35.

Is \({2}\) a solution to:

\(8x+2\)

\(=\)

\(9x-3\)

\(\wonder{=}\)

36.

Is \({{\frac{5}{7}}}\) a solution to:

\(7x^{2}-3x-{\frac{40}{7}}\)

\(=\)

\(0\)

\(\wonder{=}\)

\(0\)

37.

Is \({8}\) a solution to:

\(\left|3x+5\right|\)

\(=\)

\(32\)

\(\wonder{=}\)

\(32\)

Check a Possible Solution to an Inequality.

Check if the given number is a solution to the given inequality.

38.

Is \({-7.2}\) a solution to:

\(3x+5.2\)

\(\leq\)

\(-16.4\)

\(\wonder{\leq}\)

\(-16.4\)

39.

Is \({{\frac{2}{3}}}\) a solution to:

\(5x - {\frac{3}{4}}\)

\(\lt\)

\(7x-{\frac{25}{12}}\)

\(\wonder{\lt}\)

40.

Is \({{\frac{3}{4}}}\) a solution to:

\(4x^{2}-8x+{\frac{15}{4}}\)

\(\gt\)

\(0\)

\(\wonder{\gt}\)

\(0\)

41.

Is \({-2}\) a solution to:

\(\left|7x-2\right|\)

\(\gt\)

\(16\)

\(\wonder{\gt}\)

\(16\)

42.

Select the equations/inequalities that are linear with one variable.

\(\displaystyle 7z+9p^{2}=4\)

\(\displaystyle 2\pi r=11\pi \)

\(\displaystyle \sqrt{-6.4r-4}\leq1\)

\(\displaystyle 6+7q^{2}=18\)

\(\displaystyle 3r+3=-1\)

\(\displaystyle 7.4z\geq-2.8\)

None of the above

43.

The rental fee for a beach house is a flat \({\$350}\) plus \({\$100}\) per night. So if you stay \(n\) nights, the total is \({350+100n}\text{.}\) If the total was \({\$850}\text{,}\) then we have an equation \({350+100n}={850}\text{.}\) Did you stay \(4\) nights?

\(350+100n\)

\(=\)

\(850\)

\(\wonder{=}\)

\(850\)

44.

When a young tree was planted in your schoolâs garden, it was 6 feet tall. It grows 6/7 feet per year. After some number \(n\) of years, the tree is 24 feet tall. This gives us the equation \({6+{\frac{6}{7}}n}={24}\text{.}\) Has it been \(20\) years?

\(6+{\frac{6}{7}}n\)

\(=\)

\(24\)

\(\wonder{=}\)

\(24\)

45.

Consider a right triangle with legs of lengths \(a\) and \(b\text{,}\) and hypotenuse (the diagonal side) of length \(c\text{.}\)

A famous fact about such a triangle is that \(c=\sqrt{a^2+b^2}\text{.}\) So if one leg \(a\) is 16 inches long, and if the perimter is 34 inches long, then we have an equation \({16+b+\sqrt{16^{2}+b^{2}}}={80}\text{.}\) Is the other leg \(29\) inches long?

\(16+b+\sqrt{16^{2}+b^{2}}\)

\(=\)

\(80\)

\(\wonder{=}\)

\(80\)

SectionÂ 5: Solving One-Step Equations

Solve the Equation.

Solve the equation.

46.

\({Y-1}={11}\)

47.

\({-d}={7}\)

48.

\({j-{\frac{3}{7}}}={-{\frac{3}{8}}}\)

49.

\({p+20.7}={-15.3}\)

50.

\({\frac{v}{6.5}}={1.3}\)

51.

A convention among contractors is that steps in a staircase should have rise \(S\) and run \(N\text{,}\) both in inches, such that \(S + N = 17.5\text{.}\) (See ExampleÂ 1.1.6.) To bridge the first floor to the second floor, contractors determined the rise of each stair should be the given number of inches. Write an equation that can be used to find the run of each step. Then solve that equation and report what the run should be.

The rise is \(7.5\) inches.

SectionÂ 6: Solving One-Step Inequalities

Solve the Inequality.

Solve the inequality. Graph the solution set, and write the solution set using both interval notation and set-builder notation.

52.

\({\frac{F}{2}}\lt{6}\)

53.

\({{\frac{4}{9}}L}\leq{{\frac{4}{9}}}\)

SectionÂ 7: Algebraic Properties and Simplifying Expressions

54.

Find the multiplicative inverse of \(7\)

55.

What number is the additive identity?

Apply an Algebraic Property.

Demonstrate that you know the meanings of the various algebraic properties by applying the given algebraic property to the given expression to get a new expression.

56.

Apply associativity to \({\left(14d\right)B}\text{.}\)

57.

Apply commutativity of multiplication to \({3\mathopen{}\left(j+2\right)}\text{.}\)

58.

Apply the distributive property to \({\frac{9}{4}\mathopen{}\left(p+\frac{6}{7}\right)}\text{.}\)

Simplify.

Simplify the given expression. Ideally, you are thinking about how the properties of algebra are helping you simplify.

SectionÂ 8: Modeling with Equations and Inequalities

63.

Identify a variable you might use to represent each quantity. Then identify what units would be most appropriate.

(a)

Let be the age of a person, measured in .

(b)

Let be the distance traveled by a driver that left Portland, Oregon, bound for Boise, Idaho, measured in .

(c)

Let be the surface area of the walls of a room, measured in .

Translating English into Math.

Translate the phrase or sentence into a math expression or equation (whichever is appropriate).

64.

ten increased by a number

65.

three more than the product of eight and a number

66.

a number decreased by three fifths of itself

67.

A number divided by six is twelve.

68.

Seven less than twelve times a number is six.

Applications

Modeling with Linear Equations.

Write an equation to model the scenario. There is no need to solve the equation.

69.

When an heating oil tank is decommissioned, it is drained of its remaining oil and then filled with an inert material, such as sand. One cylindrical oil tank has a volume of \({305\ {\rm gal}}\) and is being filled with sand at a rate of \({500\ {\textstyle\frac{\rm\mathstrut gal}{\rm\mathstrut hr}}}\text{.}\) Write an equation where the solution is the amount of time, in hours, that it will take to fill the tank with sand. There is no need to solve the equation.

70.

Laney filled the gas tank in her car to \({16\ {\rm gal}}\text{.}\) When the tank reaches\({1\ {\rm gal}}\text{,}\) the low gas light will come on. On average, Laneyâs car uses \({0.045\ {\textstyle\frac{\rm\mathstrut gal}{\rm\mathstrut mi}}}\) per mile driven. How many miles will Laneyâs car be able to drive before the low gas light comes on? Write an equation to model this scenario. There is no need to solve it.

71.

A small town would like to replace its aging water treatment system. This will cost \({\$13{,}100{,}000}\text{,}\) but the town just needs \({\$1{,}310{,}000}\) up front for downpayment on a loan that will cover the rest. The town treasury has \({\$327{,}500}\) in it already for this need, and the town can gather \({\$165{,}000}\) per month from taxes. How long will it take to reach enough for downpayment on that loan? Write an equation to model this scenario. There is no need to solve it.

72.

Ricky baked a pie at \(425\,â\) and just took it out of the oven. It immediately starts to cool at a rate of \(30\,\frac{â}{\text{min}}\text{.}\) How long will it take to cool to \(225\,â\text{?}\) Write an equation to model this scenario. There is no need to solve it.

73.

At a recent trip to the casino, Tucker brought \({\$920}\) in cash. He knows he needs to hold on to \({\$80}\) in reserve to pay for dinner later. Unfortunately Tucker had rough luck and was losing money at the slot machines at an average rate of \({\$240}\) per hour. How long was Tucker gambling before he had to stop? Write an equation to model this scenario. There is no need to solve it.

74.

Alyssaâs current annual salary as a dental hygienist is \({\$59{,}751}\text{.}\) This is with a raise of \({5.2\%}\) over last yearâs salary. What was her salary last year? Set up an equation to answer this question. There is no need to solve it.

75.

One year, the median rent for a one-bedroom apartment in a city was reported to be \({\$1{,}200}\text{.}\) This was reported to be an increase of \({2.8\%}\) over the previous year. Based on this reporting, what was the median rent for of a one-bedroom apartment the previous year? Set up an equation to answer this question. There is no need to solve it.

Modeling with Linear Inequalities.

Write an inequality to model the scenario. There is no need to solve the inequality.

76.

Elâs maximum lung capacity is \({7.2\ {\rm L}}\text{.}\) If their lungs are full and they exhale at a rate of \({0.5\ {\textstyle\frac{\rm\mathstrut L}{\rm\mathstrut s}}}\text{,}\) write an inequality where the solution set is the possible times when they still have at least \({0.6\ {\rm L}}\) of air left in their lungs. There is no need to solve it.

77.

The final bill at a restaurant one night was \({\$84}\text{,}\) including a tip that was at least \({17\%}\text{.}\) What could the bill have been before the tip was added? Set up an inequality where the solution set represents the possible amounts that the original bill might have been. There is no need to solve it.