1.5: Exercises

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Page ID: 48951

Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

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Exercise $\PageIndex{1}$

Give examples of numbers that are

natural numbers
integers
integers but not natural numbers
rational numbers
real numbers
rational numbers but not integers

Answer

$2, 3, 5$
$−3, 0, 6$
$−3, −4, 0$
$\dfrac{2}{3}, \dfrac{-4}{7}, 8$
$\sqrt{5}, \pi, \sqrt[3]{31}$
$\dfrac{1}{2}, \dfrac{2}{5}, 0.75$

Exercise $\PageIndex{2}$

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

$\dfrac 7 3$
$-5$
$0$
$17,000$
$\dfrac{12}{4}$
$\sqrt{7}$
$\sqrt{25}$

Answer

rational
integer, rational
integer, rational
natural, integer, rational
natural, integer, rational
irrational
natural, integer, rational

All of the given numbers are real numbers

Exercise $\PageIndex{3}$

Evaluate the following absolute value expressions:

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

Answer

$8$
$10$
$99$
$-3$
$-2$
$\sqrt{6}$
$7$
$7$
$5.4$
$\dfrac{2}{3}$
$\dfrac{5}{2}$
$-2$

Exercise $\PageIndex{4}$

Solve for $x$:

$|x|=8$
$|x|=0$
$|x|=-3$
$|x+3|=10$
$|2x+5|=9$
$|2-5x|=22$
$|4x|=-8$
$|-7x-3|=0$
$|4-4x|=44$
$-2\cdot |2-3x|=-12$
$5+|2x+7|=14$
$-|-8-2x|=-12$

Answer

$S=\{-8,8\}$
$S=\{0\}$
$S=\{\}$
$S=\{-13,7\}$
$S=\{-7,2\}$
$S=\left\{-4, \dfrac{24}{5}\right\}$
$S=\{\}$
$S=\left\{\dfrac{-3}{7}\right\}$
$S=\{-10,12\}$
$S=\left\{\dfrac{-4}{3}, \dfrac{8}{3}\right\}$
$S=\{-8,1\}$
$S=\{-10,2\}$

Exercise $\PageIndex{5}$

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

$|x|=8$
$|x|=0$
$|x|=-3$
$|x-4|=2$
$|x+5|=9$
$|2-x|=5$

Answer

$S=\{-8,8\}$
$S=\{0\}$
$S=\{\}$
$S=\{2,6\}$
$S=\{-14,4\}$
$S=\{-3,7\}$

Exercise $\PageIndex{6}$

Complete the table.

Inequality notation	Number line	Interval notation
$2\leq x< 5$
$x\leq 3$
	$clipboard_ee36fbb8a1140f74a4f63d1e8e3ff679a.png$
	$clipboard_e6c534d4d0379a6b2fddb24f85dc1f57e.png$
		$[-2,6]$
		$(-\infty,0)$
	$clipboard_ee530401a798598d9e9a7c8efd7988e37.png$
$5< x\leq \sqrt{30}$
		$\left(\dfrac{13}{7},\pi \right )$

Answer

Inequality notation	Number line	Interval notation
$2\leq x< 5$	$clipboard_ecc556141ef9c6812ed11f16885af51e9.png$	$[2,5)$
$x\leq 3$	$clipboard_e7d2ae6b9269b96fcb9abdf5ab7067ab5.png$	$(-\infty, 3]$
$12<x \leq 17$	$clipboard_ee36fbb8a1140f74a4f63d1e8e3ff679a.png$	$(12,17]$
$x< -2$	$clipboard_e6c534d4d0379a6b2fddb24f85dc1f57e.png$	$(-\infty,-2)$
$-2 \leq x \leq 6$	$clipboard_e98502b65d85cdc5539d27cf17265fc06.png$	$[-2,6]$
$x< 0$	$clipboard_ecc1e5c8860f45a8764dba047a1cdfcb3.png$	$(-\infty,0)$
$4.5 \leq x$	$clipboard_ee530401a798598d9e9a7c8efd7988e37.png$	$[4.5, \infty)$
$5< x\leq \sqrt{30}$	$clipboard_ea672b8a6eaaa02d6a68eec34566831e7.png$	$(5, \sqrt{30}]$
$\dfrac{13}{7}<x<\pi$	$clipboard_edc9edf9b56a08bf45919a6fc47142c9a.png$	$\left(\dfrac{13}{7},\pi \right )$

Exercise $\PageIndex{7}$

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

$|x-4|\leq 7$
$|x-4|\geq 7$
$|x-4|> 7$
$|2x+7|\leq 13$
$|-2-4x|>8$
$|4x+2|<17$
$|15-3x|\geq 6$
$\left|5x-\dfrac 4 3 \right|>\dfrac 2 3$
$\left| \sqrt{2}x-\sqrt{2}\right|\leq \sqrt{8}$
$|2x+3|<-5$
$|5+5x|\geq -2$
$|5+5x|> 0$

Answer

$S=[-3,11]$
$S=(-\infty,-3] \cup[11, \infty)$
$S=(-\infty,-3) \cup(11, \infty)$
$S=[-10,3],$
$S=\left(-\infty,-\dfrac{5}{2}\right) \cup\left(\dfrac{3}{2}, \infty\right)$
$S=\left(\dfrac{-19}{4}, \dfrac{15}{4}\right)$
$S=(-\infty, 3] \cup[7, \infty)$
$S=\left(-\infty, \dfrac{2}{15}\right) \cup\left(\dfrac{2}{5}, \infty\right)$
$S=[-1,3]$
$S=\{\}$
$S=(-\infty, \infty)=\mathbb{R}$
$S=(-\infty,-1) \cup(-1, \infty)=\mathbb{R}-\{-1\}$

Inequality notation	Number line	Interval notation
\(2\leq x< 5\)	$clipboard_ecc556141ef9c6812ed11f16885af51e9.png$	\([2,5)\)
\(x\leq 3\)	$clipboard_e7d2ae6b9269b96fcb9abdf5ab7067ab5.png$	\((-\infty, 3]\)
\(12<x \leq 17\)	$clipboard_ee36fbb8a1140f74a4f63d1e8e3ff679a.png$	\((12,17]\)
\(x< -2\)	$clipboard_e6c534d4d0379a6b2fddb24f85dc1f57e.png$	\((-\infty,-2)\)
\(-2 \leq x \leq 6\)	$clipboard_e98502b65d85cdc5539d27cf17265fc06.png$	\([-2,6]\)
\(x< 0\)	$clipboard_ecc1e5c8860f45a8764dba047a1cdfcb3.png$	\((-\infty,0)\)
\(4.5 \leq x\)	$clipboard_ee530401a798598d9e9a7c8efd7988e37.png$	\([4.5, \infty)\)
\(5< x\leq \sqrt{30}\)	$clipboard_ea672b8a6eaaa02d6a68eec34566831e7.png$	\((5, \sqrt{30}]\)
\(\dfrac{13}{7}<x<\pi\)	$clipboard_edc9edf9b56a08bf45919a6fc47142c9a.png$	\(\left(\dfrac{13}{7},\pi \right )\)

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