# 1.5: Exercises

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## Exercise $$\PageIndex{1}$$

Give examples of numbers that are

1. natural numbers
2. integers
3. integers but not natural numbers
4. rational numbers
5. real numbers
6. rational numbers but not integers
1. $$2, 3, 5$$
2. $$−3, 0, 6$$
3. $$−3, −4, 0$$
4. $$\dfrac{2}{3}, \dfrac{-4}{7}, 8$$
5. $$\sqrt{5}, \pi, \sqrt[3]{31}$$
6. $$\dfrac{1}{2}, \dfrac{2}{5}, 0.75$$

## Exercise $$\PageIndex{2}$$

Which of the following numbers are natural numbers, integers, rational numbers, or real numbers? Which of these numbers are irrational?

1. $$\dfrac 7 3$$
2. $$-5$$
3. $$0$$
4. $$17,000$$
5. $$\dfrac{12}{4}$$
6. $$\sqrt{7}$$
7. $$\sqrt{25}$$
1. rational
2. integer, rational
3. integer, rational
4. natural, integer, rational
5. natural, integer, rational
6. irrational
7. natural, integer, rational

All of the given numbers are real numbers

## Exercise $$\PageIndex{3}$$

Evaluate the following absolute value expressions:

1. $$|-8|$$
2. $$|10|$$
3. $$|-99|$$
4. $$-|3|$$
5. $$-|-2|$$
6. $$|-\sqrt{6}|$$
7. $$|3+4|$$
8. $$|2-9|$$
9. $$|-5.4|$$
10. $$\left|-\dfrac{2}{3}\right|$$
11. $$\left|\dfrac{5}{-2}\right|$$
12. $$-\left|-\dfrac{-6}{-3}\right|$$
1. $$8$$
2. $$10$$
3. $$99$$
4. $$-3$$
5. $$-2$$
6. $$\sqrt{6}$$
7. $$7$$
8. $$7$$
9. $$5.4$$
10. $$\dfrac{2}{3}$$
11. $$\dfrac{5}{2}$$
12. $$-2$$

## Exercise $$\PageIndex{4}$$

Solve for $$x$$:

1. $$|x|=8$$
2. $$|x|=0$$
3. $$|x|=-3$$
4. $$|x+3|=10$$
5. $$|2x+5|=9$$
6. $$|2-5x|=22$$
7. $$|4x|=-8$$
8. $$|-7x-3|=0$$
9. $$|4-4x|=44$$
10. $$-2\cdot |2-3x|=-12$$
11. $$5+|2x+7|=14$$
12. $$-|-8-2x|=-12$$
1. $$S=\{-8,8\}$$
2. $$S=\{0\}$$
3. $$S=\{\}$$
4. $$S=\{-13,7\}$$
5. $$S=\{-7,2\}$$
6. $$S=\left\{-4, \dfrac{24}{5}\right\}$$
7. $$S=\{\}$$
8. $$S=\left\{\dfrac{-3}{7}\right\}$$
9. $$S=\{-10,12\}$$
10. $$S=\left\{\dfrac{-4}{3}, \dfrac{8}{3}\right\}$$
11. $$S=\{-8,1\}$$
12. $$S=\{-10,2\}$$

## Exercise $$\PageIndex{5}$$

Solve for $$x$$ using the geometric interpretation of the absolute value:

1. $$|x|=8$$
2. $$|x|=0$$
3. $$|x|=-3$$
4. $$|x-4|=2$$
5. $$|x+5|=9$$
6. $$|2-x|=5$$
1. $$S=\{-8,8\}$$
2. $$S=\{0\}$$
3. $$S=\{\}$$
4. $$S=\{2,6\}$$
5. $$S=\{-14,4\}$$
6. $$S=\{-3,7\}$$

## Exercise $$\PageIndex{6}$$

Complete the table.

Inequality notation Number line Interval notation
$$2\leq x< 5$$
$$x\leq 3$$

$$[-2,6]$$
$$(-\infty,0)$$

$$5< x\leq \sqrt{30}$$
$$\left(\dfrac{13}{7},\pi \right )$$
Inequality notation Number line Interval notation
$$2\leq x< 5$$ $$[2,5)$$
$$x\leq 3$$ $$(-\infty, 3]$$
$$12<x \leq 17$$ $$(12,17]$$
$$x< -2$$ $$(-\infty,-2)$$
$$-2 \leq x \leq 6$$ $$[-2,6]$$
$$x< 0$$ $$(-\infty,0)$$
$$4.5 \leq x$$ $$[4.5, \infty)$$
$$5< x\leq \sqrt{30}$$ $$(5, \sqrt{30}]$$
$$\dfrac{13}{7}<x<\pi$$ $$\left(\dfrac{13}{7},\pi \right )$$

## Exercise $$\PageIndex{7}$$

Solve for $$x$$ and write the solution in interval notation.

1. $$|x-4|\leq 7$$
2. $$|x-4|\geq 7$$
3. $$|x-4|> 7$$
4. $$|2x+7|\leq 13$$
5. $$|-2-4x|>8$$
6. $$|4x+2|<17$$
7. $$|15-3x|\geq 6$$
8. $$\left|5x-\dfrac 4 3 \right|>\dfrac 2 3$$
9. $$\left| \sqrt{2}x-\sqrt{2}\right|\leq \sqrt{8}$$
10. $$|2x+3|<-5$$
11. $$|5+5x|\geq -2$$
12. $$|5+5x|> 0$$
1. $$S=[-3,11]$$
2. $$S=(-\infty,-3] \cup[11, \infty)$$
3. $$S=(-\infty,-3) \cup(11, \infty)$$
4. $$S=[-10,3],$$
5. $$S=\left(-\infty,-\dfrac{5}{2}\right) \cup\left(\dfrac{3}{2}, \infty\right)$$
6. $$S=\left(\dfrac{-19}{4}, \dfrac{15}{4}\right)$$
7. $$S=(-\infty, 3] \cup[7, \infty)$$
8. $$S=\left(-\infty, \dfrac{2}{15}\right) \cup\left(\dfrac{2}{5}, \infty\right)$$
9. $$S=[-1,3]$$
10. $$S=\{\}$$
11. $$S=(-\infty, \infty)=\mathbb{R}$$
12. $$S=(-\infty,-1) \cup(-1, \infty)=\mathbb{R}-\{-1\}$$

This page titled 1.5: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.