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4-105.

Think of absolute value as a statement about distance as you answer the questions below. What values of $x$ in parts (a) through (d) make each equation true?

1. $| x - 7 | = 50$

What numbers are a distance of $50$ from 7? Picture this on a number line.

$x = 57$, $-43$

1. $| x + 7 | = 50$

See part (a).

1. $| 10 - x | = 12$

See part (a).

1. $| 2 x + 1 | = - 3$

See part (a)

Remember absolute values cannot be negative.

1. What mathematical operation is best used for calculating the distance between two numbers? In other words, if you want to know the distance from $42$ to $117$, what arithmetic expression represents that distance?

Subtraction, $117 - 42$

2. Suppose you want to write an equation to represent the statement, “The distance between a number and $47$ is $21$.” You would not know whether to write $x - 47 = 21$ or $47 - x = 21$. Absolute value equations can allow you to write a correct expression without knowing which value is larger. Write two absolute value equations that mean “the distance between a number and $47$ is $21$.”

$|x - 47| = 21$

3. Write and solve an absolute value equation that says each of the following:

See the answer to part (f).

1. “The distance between $x$ and $4$ is $12$.”

$|4 - x| = 12$ or $|x - 4| = 12$

1. “The distance between $x$ and $−9$ is $15$.”