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Absolute value answers the question of "how far," and not "which way." The phrase "how far" implies "length" and length is always a nonnegative quantity . Thus, the absolute value of a number is a nonnegative number.
Determine each value.
$\mid 4\mid =4$
$\mid -4\mid =4$
$\mid 0\mid =0$
$-\mid 5\mid =-5$ . The quantity on the left side of the equal sign is read as "negative the absolute value of 5." The absolute value of 5 is 5. Hence, negative the absolute value of 5 is -5.
$-\mid -3\mid =-3$ . The quantity on the left side of the equal sign is read as "negative the absolute value of -3." The absolute value of -3 is 3. Hence, negative the absolute value of -3 is $-\left(3\right)=-3$ .
By reasoning geometrically, determine each absolute value.
$\mid 7\mid $
7
$\mid -3\mid $
3
$\mid \text{12}\mid $
12
$\mid 0\mid $
0
$-\mid 9\mid $
-9
$-\mid -6\mid $
-6
From the problems in [link] , we can suggest the following algebraic definition of absolute value. Note that the definition has two parts.
The algebraic definition takes into account the fact that the number $a$ could be either positive or zero $\left(a\ge 0\right)$ or negative $\left(a<0\right)$ .
Use the algebraic definition of absolute value to find the following values.
$\mid 8\mid $ . The number enclosed within the absolute value bars is a nonnegative number, so the upper part of the definition applies. This part says that the absolute value of 8 is 8 itself.
$\mid 8\mid =8$
$\mid -3\mid $ . The number enclosed within absolute value bars is a negative number, so the lower part of the definition applies. This part says that the absolute value of -3 is the opposite of -3, which is $-\left(-3\right)$ . By the definition of absolute value and the double-negative property,
$\mid -3\mid =-\left(-3\right)=3$
Use the algebraic definition of absolute value to find the following values.
$\mid 7\mid $
7
$\mid 9\mid $
9
$\mid -\text{12}\mid $
12
$\mid -5\mid $
5
$-\mid 8\mid $
-8
$-\mid 1\mid $
-1
$-\mid -\text{52}\mid $
-52
$-\mid -31\mid $
-31
Determine each of the values.
$\mid 5\mid $
5
$\mid 3\mid $
$\mid 6\mid $
6
$\mid -9\mid $
$\mid -1\mid $
1
$\mid -4\mid $
$-\mid 3\mid $
-3
$-\mid 7\mid $
$-\mid -14\mid $
-14
$\mid 0\mid $
$\mid -\text{26}\mid $
26
$-\mid -\text{26}\mid $
$-\left(-\mid 4\mid \right)$
4
$-\left(-\mid 2\mid \right)$
$-\left(-\mid -6\mid \right)$
6
$-\left(-\mid -\text{42}\mid \right)$
$\mid 5\mid -\mid -2\mid $
3
${\mid -2\mid}^{3}$
$\mid -\left(2\cdot 3\right)\mid $
6
$\mid -2\mid -\mid -9\mid $
${\left(\mid -6\mid +\mid 4\mid \right)}^{2}$
100
${\left(\mid -1\mid -\mid 1\mid \right)}^{3}$
${\left(\mid 4\mid +\mid -6\mid \right)}^{2}-{\left(\mid -2\mid \right)}^{3}$
92
$-{\left[\left|-10\right|-6\right]}^{2}$
$-{\{-{\left[-\mid -4\mid +\mid -3\mid \right]}^{3}\}}^{2}$
-1
A Mission Control Officer at Cape Canaveral makes the statement “lift-off, T minus 50 seconds.” How long is it before lift-off?
Due to a slowdown in the industry, a Silicon Valley computer company finds itself in debt $2,400,000. Use absolute value notation to describe this company’s debt.
$-\$\mid -\mathrm{2,}\text{400},\text{000}\mid $
A particular machine is set correctly if upon action its meter reads 0. One particular machine has a meter reading of $-1.6$ upon action. How far is this machine off its correct setting?
( [link] ) Find the sum: $\frac{9}{\text{70}}+\frac{5}{\text{21}}+\frac{8}{\text{15}}$ .
$\frac{9}{\text{10}}$
( [link] ) Find the value of $\frac{\frac{3}{\text{10}}+\frac{4}{\text{12}}}{\frac{\text{19}}{\text{20}}}$ .
( [link] ) Convert $3\text{.}2\frac{3}{5}$ to a fraction.
$3\frac{\text{13}}{\text{50}}\text{or}\frac{\text{163}}{\text{50}}$
( [link] ) The ratio of acid to water in a solution is $\frac{3}{8}$ . How many mL of acid are there in a solution that contain 112 mL of water?
( [link] ) Find the value of $-6-(-8)$ .
2
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