1.5: Exercises
- Page ID
- 48951
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Give examples of numbers that are
- natural numbers
- integers
- integers but not natural numbers
- rational numbers
- real numbers
- rational numbers but not integers
- Answer
-
- \(2, 3, 5\)
- \(−3, 0, 6\)
- \(−3, −4, 0\)
- \(\dfrac{2}{3}, \dfrac{-4}{7}, 8\)
- \(\sqrt{5}, \pi, \sqrt[3]{31}\)
- \(\dfrac{1}{2}, \dfrac{2}{5}, 0.75\)
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
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
-
- \(8\)
- \(10\)
- \(99\)
- \(-3\)
- \(-2\)
- \(\sqrt{6}\)
- \(7\)
- \(7\)
- \(5.4\)
- \(\dfrac{2}{3}\)
- \(\dfrac{5}{2}\)
- \(-2\)
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
-
- \(S=\{-8,8\}\)
- \(S=\{0\}\)
- \(S=\{\}\)
- \(S=\{-13,7\}\)
- \(S=\{-7,2\}\)
- \(S=\left\{-4, \dfrac{24}{5}\right\}\)
- \(S=\{\}\)
- \(S=\left\{\dfrac{-3}{7}\right\}\)
- \(S=\{-10,12\}\)
- \(S=\left\{\dfrac{-4}{3}, \dfrac{8}{3}\right\}\)
- \(S=\{-8,1\}\)
- \(S=\{-10,2\}\)
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
-
- \(S=\{-8,8\}\)
- \(S=\{0\}\)
- \(S=\{\}\)
- \(S=\{2,6\}\)
- \(S=\{-14,4\}\)
- \(S=\{-3,7\}\)
Complete the table.
Inequality notation | Number line | Interval notation |
---|---|---|
\(2\leq x< 5\) | ||
\(x\leq 3\) | ||
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\([-2,6]\) | ||
\((-\infty,0)\) | ||
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\(5< x\leq \sqrt{30}\) | ||
\(\left(\dfrac{13}{7},\pi \right )\) |
- Answer
-
Inequality notation Number line Interval notation \(2\leq x< 5\) \([2,5)\) \(x\leq 3\) \((-\infty, 3]\) \(12<x \leq 17\) \((12,17]\) \(x< -2\) \((-\infty,-2)\) \(-2 \leq x \leq 6\) \([-2,6]\) \(x< 0\) \((-\infty,0)\) \(4.5 \leq x\) \([4.5, \infty)\) \(5< x\leq \sqrt{30}\) \((5, \sqrt{30}]\) \(\dfrac{13}{7}<x<\pi\) \(\left(\dfrac{13}{7},\pi \right )\)
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
-
- \(S=[-3,11]\)
- \(S=(-\infty,-3] \cup[11, \infty)\)
- \(S=(-\infty,-3) \cup(11, \infty)\)
- \(S=[-10,3],\)
- \(S=\left(-\infty,-\dfrac{5}{2}\right) \cup\left(\dfrac{3}{2}, \infty\right)\)
- \(S=\left(\dfrac{-19}{4}, \dfrac{15}{4}\right)\)
- \(S=(-\infty, 3] \cup[7, \infty)\)
- \(S=\left(-\infty, \dfrac{2}{15}\right) \cup\left(\dfrac{2}{5}, \infty\right)\)
- \(S=[-1,3]\)
- \(S=\{\}\)
- \(S=(-\infty, \infty)=\mathbb{R}\)
- \(S=(-\infty,-1) \cup(-1, \infty)=\mathbb{R}-\{-1\}\)