In the last section we studied some linear models, and in particular we looked at graphs and equations that described those models. In this section we review some techniques and terminology related to equations, inequalities, and their graphs.
A linear equation has no powers of the variable other than 1. It has at most one solution. We find the solution by transforming the equation into a simpler equivalent equation whose solution is obvious.
Although a linear equation can have at most one solution, a linear inequality can have many solutions. For example, complete the table of values for the expression \(5 β 2x\text{:}\)
We begin by isolating the term containing the variable, just as we do when solving a linear equation. We subtract 5 from both sides to obtain
\begin{equation*}
-2x \lt -3
\end{equation*}
Then we divide both sides by \(β2\) to find
\begin{equation*}
x \alert{\gt} \frac{-3}{-2}=\frac{3}{2} ~~~~~~~~~~~~~~~~ \blert{\text{Reverse the direction of the inequality.}}
\end{equation*}
Any value of \(x\) greater than \(\frac{3}{2}\) is a solution of the inequality. We write the solutions as \(x \gt \frac{3}{2}\text{.}\) Because we cannot list all of these solutions, we often illustrate them as a graph on a number line, as shown below.
Other than the rule stated in the box above, the rules for solving a linear inequality are the same as the rules for solving a linear equation.\(~~~\alert{\text{[TK]}}~~\)
has many solutions. Each solution consists of an ordered pair of values, one for \(x\) and one for \(y\text{,}\) that together satisfy the equation (make the equation true.)
You might think it would be difficult to find all the solutions of an equation, but for a linear equation \(Ax + By = C\text{,}\) we can at least illustrate the solutions: all the solutions lie on a straight line. (Later on we can prove that this is true.)
The point \((0, 24)\) lies on the graph, so \(x = 0\text{,}\)\(y = 24\) is a solution of the equation. Thus, \(A \cdot 0 + B \cdot 24 = C\text{,}\) or \(24B = C\) is a true statement.
On the grid below, plot the points you found in Practice 3. All the points should lie on a straight line; draw the line with a ruler or straightedge. Which of the following points lie on the graph?
Subsection1.2.4Graphical Solution of Equations and Inequalities
Here is a clever way to solve an equation in one variable by using a graph. Suppose we would like to solve the equation \(~150 = 285 - 15x\text{.}\) We start by looking at the graph of \(~y = 285 - 15x\text{.}\)
Compare the two equations in the problem. In the equation we want to solve, \(y\) has been replaced by \(\alert{150}\text{.}\) We begin by locating the point \(P\) on the graph for which \(y = 150\text{.}\)
Next we find the \(x\)-coordinate of point \(P\) by drawing an imaginary line from \(P\) straight down to the \(x\)-axis. The \(x\)-coordinate of \(P\) is \(x = 9\text{.}\)
The relationship between an equation and its graph is an important one. For the previous example, make sure you understand that the following three statements are equivalent.
We can also use graphs to solve inequalities. Consider the inequality
\begin{equation*}
285-15x \ge 150
\end{equation*}
To solve this inequality means to find all values of \(x\) that make the expression \(285 -15x\) greater than or equal to 150. We could begin by trying some values of \(x\text{.}\) Here is a table obtained by evaluating \(285 β 15x\text{.}\)
From the table, we see that values of \(x\) less than or equal to 8 are solutions of the inequality, but we have not checked all possible \(x\)-values. We can get a more complete picture from a graph.
We look for points on the graph with \(y\)-coordinates greater than or equal to 150. These points are shown in color. Which \(x\)-values produced these points?
We can read the \(x\)-coordinates by dropping straight down to the \(x\)-axis, as shown by the arows. For example, the \(x\)-value corresponding to \(y=150\) is \(x=9\text{.}\) For larger values of \(285-15x\text{,}\) we must choose \(x\)-values less than 9. Thus, all values of \(x\) less than or equal to 9 are solutions, as shown on the \(x\)-axis.
We can use a graphing utility to graph equations if they are written in the form \(~y = (\text{expression in }x)~\text{.}\) First, letβs review how to solve an equation for \(y\) in terms of \(x\text{.}\)
Choosing a graphing window corresponds to drawing the \(x\)- and \(y\)-axes and marking a scale on each axis when we graph by hand. The standard graphing window displays values from \(-10\) to \(10\) on both axes. We can start with this window and then adjust it if necessary.
We press TRACE, and a "bug" begins flashing on the display. The coordinates of the bug appear at the bottom of the display, as shown in the figure. We use the left and right arrows to move the bug along the graph. You can check that the coordinates of the point \((2, -10.6)\) do satisfy the equation \(y = -2.6x - 5.4\text{.}\)
Kieranβs resting blood pressure, in mm Hg, is 120, and it rises by 6 mm for each minute he jogs on a treadmill programmed to increase the level of intensity at a steady rate.
Find a formula for Kieranβs blood pressure, \(p\text{,}\) in terms of time, \(t\text{.}\)
When Francine is at rest, her cardiac output is 5 liters per minute. The output increases by 3 liters per minute for each minute she spends on a cycling machine with increasing intensity.
Find a formula for Francineβs cardiac output, \(c\text{,}\) in terms of time, \(t\text{.}\)