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Section4.10Using 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 FigureĀ 4.10.1, 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 \([-4.5,3]\text{.}\)

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Figure4.10.1\(y=g(x)\)

In FigureĀ 4.10.2, 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 \((-\infty,-3) \cup (0,6]\text{.}\)

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Figure4.10.2\(y=g(x)\)

Subsection4.10.1Exercises

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 FigureĀ 4.10.3. State the solution set using both set-builder notation and interval notation.

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Figure4.10.3\(y=f(x)\)

Solution

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

The solution set is \((-2,0) \cup (2,4)\text{.}\)

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Figure4.10.4\(f(x) \lt 2\)
2

Determine the solution set to \(f(x) \geq 2\) based upon the function \(f\) shown in FigureĀ 4.10.5. State the solution set using both set-builder notation and interval notation.

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Figure4.10.5\(y=f(x)\)

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 \([-4,-2] \cup [0,2] \cup [4,5)\text{.}\)

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Figure4.10.6\(f(x) \ge 2\)
3

Determine the solution set to \(f(x) \lt -1\) based upon the function \(f\) shown in FigureĀ 4.10.7. State the solution set using both set-builder notation and interval notation.

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Figure4.10.7\(y=f(x)\)

Solution

The solution set is \(\{\}\text{.}\)

The solution set is \(\emptyset\text{.}\)

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Figure4.10.8\(f(x) \lt -1\)

4

Determine the solution set to \(f(x) \leq 5\) based upon the function \(f\) shown in FigureĀ 4.10.9. State the solution set using both set-builder notation and interval notation.

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Figure4.10.9\(y=f(x)\)

Solution

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

The solution set is \((-4,5)\text{.}\)

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Figure4.10.10\(f(x) \le 5\)
5

Determine the solution set to \(g(x) \geq -2\) based upon the function \(g\) shown in FigureĀ 4.10.11. State the solution set using both set-builder notation and interval notation.

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Figure4.10.11\(y=g(x)\)

Solution

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

The solution set is \([-2,3]\text{.}\)

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Figure4.10.12\(g(x) \ge -2\)
6

Determine the solution set to \(g(x) \leq 3\) based upon the function \(g\) shown in FigureĀ 4.10.13. State the solution set using both set-builder notation and interval notation.

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Figure4.10.13\(y=g(x)\)

Solution

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

The solution set is \((-6,-1) \cup \left[\frac{1}{2},\infty\right)\text{.}\)

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Figure4.10.14\(g(x) \le 3\)
7

Determine the solution set to \(g(x) \lt -6\) based upon the function \(g\) shown in FigureĀ 4.10.15. State the solution set using both set-builder notation and interval notation.

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Figure4.10.15\(y=g(x)\)

Solution

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

The solution set is \((5,\infty)\text{.}\)

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Figure4.10.16\(g(x) \lt -6\)

8

Determine the solution set to \(g(x) \gt -6\) based upon the function \(g\) shown in FigureĀ 4.10.17. State the solution set using both set-builder notation and interval notation.

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Figure4.10.17\(y=g(x)\)

Solution

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

The solution set is \((-6,-4) \cup (-4,5)\text{.}\)

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Figure4.10.18\(g(x) \gt -6\)
9

Determine the solution set to \(f(x) \geq g(x)\) based upon the functions \(f\) (piecewise-linear) and \(g\) (parabolic) shown in FigureĀ 4.10.19. State the solution set using both set-builder notation and interval notation.

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Figure4.10.19\(\highlightr{y=f(x)}\) and \(\highlighty{y=g(x)}\)

Solution

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

The solution set is \([-5,1]\text{.}\)

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Figure4.10.20\(\highlightr{f(x)} \ge \highlighty{g(x)}\)
10

Determine the solution set to \(g(x) \gt f(x)\) based upon the functions \(f\) (piecewise-linear) and \(g\) (parabolic) shown in FigureĀ 4.10.21. State the solution set using both set-builder notation and interval notation.

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Figure4.10.21\(\highlightr{y=f(x)}\) 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 \((-\infty,-5) \cup (1,\infty)\text{.}\)

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Figure4.10.22\(\highlighty{g(x)} \gt \highlightr{f(x)}\)