## Section5.11Using Function Graphs to Determine Solution Sets to Inequalities

When solving inequalities involving functions that are presented in graphical form, we follow a two-step process. We first use the function to identify the points on the curve whose $-y$-coordinates satisfy the property implied by the inequality statement. We then identify the $x$-coordinates of those points, which collectively make up the solution set to the inequality.

In Figure 5.11.1, I've indicated all of the points on the function named $g$ that have $y$-coordinates greater than or equal to $1\text{.}$ I've also marked off the portion of the $x$-axis over which these points lie. Since the $y$-coordinates of the points are the values of $g(x)\text{,}$ we can infer from this that the solution set to the inequality $g(x) \geq 1$ is $[-4.5,3]\text{.}$ Figure 5.11.1. $y=g(x)$

In Figure 5.11.2, I've indicated all of the points on the function named $g$ that have $y$-coordinates less than to $44\text{.}$ I've also marked off the portion of the $x$-axis over which these points lie. Since the $y$-coordinates of the points are the values of $g(x)\text{,}$ we can infer from this that the solution set to the inequality $g(x) \lt 4$ is $(-\infty,-3) \cup (0,6]\text{.}$ Figure 5.11.2. $y=g(x)$

You can use Figure 5.11.3 and Figure 5.11.4 to investigate these idea in an interactive manner.

### ExercisesExercises

Determine the solution set to each stated inequality.

###### 1.

Determine the solution set to $f(x) \lt 2$ based upon the function $f$ shown in Figure 5.11.5. State the solution set using both set-builder notation and interval notation. Figure 5.11.5. $y=f(x)$

Solution

The solution set is $\{x \mid -2 \lt x \lt 0 \text{ or } 2 \lt x \lt 4\}\text{.}$

The solution set is $(-2,0) \cup (2,4)\text{.}$ Figure 5.11.6. $f(x) \lt 2$
###### 2.

Determine the solution set to $f(x) \geq 2$ based upon the function $f$ shown in Figure 5.11.7. State the solution set using both set-builder notation and interval notation. Figure 5.11.7. $y=f(x)$

Solution

The solution set is $\{x \mid -4 \leq x \leq -2 \text{ or } 0 \leq x \leq 2 \text{ or } 4 \leq x \lt 5\}\text{.}$

The solution set is $[-4,-2] \cup [0,2] \cup [4,5)\text{.}$ Figure 5.11.8. $f(x) \ge 2$
###### 3.

Determine the solution set to $f(x) \lt -1$ based upon the function $f$ shown in Figure 5.11.9. State the solution set using both set-builder notation and interval notation. Figure 5.11.9. $y=f(x)$

Solution

The solution set is $\{\}\text{.}$

The solution set is $\emptyset\text{.}$ Figure 5.11.10. $f(x) \lt -1$

###### 4.

Determine the solution set to $f(x) \leq 5$ based upon the function $f$ shown in Figure 5.11.11. State the solution set using both set-builder notation and interval notation. Figure 5.11.11. $y=f(x)$

Solution

The solution set is $\{x \mid -4 \leq x \lt 5\}\text{.}$

The solution set is $(-4,5)\text{.}$ Figure 5.11.12. $f(x) \le 5$
###### 5.

Determine the solution set to $g(x) \geq -2$ based upon the function $g$ shown in Figure 5.11.13. State the solution set using both set-builder notation and interval notation. Figure 5.11.13. $y=g(x)$

Solution

The solution set is $\{-2 \leq x \leq 3\}\text{.}$

The solution set is $[-2,3]\text{.}$ Figure 5.11.14. $g(x) \ge -2$
###### 6.

Determine the solution set to $g(x) \leq 3$ based upon the function $g$ shown in Figure 5.11.15. State the solution set using both set-builder notation and interval notation. Figure 5.11.15. $y=g(x)$

Solution

The solution set is $\{x \mid -6 \lt x \lt 2 \text{ or } x \geq \frac{1}{2}\}\text{.}$

The solution set is $(-6,-1) \cup \left[\frac{1}{2},\infty\right)\text{.}$ Figure 5.11.16. $g(x) \le 3$
###### 7.

Determine the solution set to $g(x) \lt -6$ based upon the function $g$ shown in Figure 5.11.17. State the solution set using both set-builder notation and interval notation. Figure 5.11.17. $y=g(x)$

Solution

The solution set is $\{x \mid x \gt 5\}\text{.}$

The solution set is $(5,\infty)\text{.}$ Figure 5.11.18. $g(x) \lt -6$

###### 8.

Determine the solution set to $g(x) \gt -6$ based upon the function $g$ shown in Figure 5.11.19. State the solution set using both set-builder notation and interval notation. Figure 5.11.19. $y=g(x)$

Solution

The solution set is $\{x \mid -6 \lt x \lt -4 \text{ or } -4 \lt x \lt 5\}\text{.}$

The solution set is $(-6,-4) \cup (-4,5)\text{.}$ Figure 5.11.20. $g(x) \gt -6$
###### 9.

Determine the solution set to $f(x) \geq g(x)$ based upon the functions $f$ (piecewise-linear) and $g$ (parabolic) shown in Figure 5.11.21. State the solution set using both set-builder notation and interval notation. Figure 5.11.21. $\highlightr{y=f(x)}$ and $\highlighty{y=g(x)}$

Solution

The solution set is $\{x \mid -5 \leq x \leq 1\}\text{.}$

The solution set is $[-5,1]\text{.}$ Figure 5.11.22. $\highlightr{f(x)} \ge \highlighty{g(x)}$
###### 10.

Determine the solution set to $g(x) \gt f(x)$ based upon the functions $f$ (piecewise-linear) and $g$ (parabolic) shown in Figure 5.11.23. State the solution set using both set-builder notation and interval notation. Figure 5.11.23. $\highlightr{y=f(x)}$ and $\highlighty{y=g(x)}$

Solution

The solution set is $\{x \mid x \lt -5 \text{ or } x \gt 1\}\text{.}$

The solution set is $(-\infty,-5) \cup (1,\infty)\text{.}$ Figure 5.11.24. $\highlighty{g(x)} \gt \highlightr{f(x)}$