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Solve for $r$ . $|r - 8| \geq 8$ Write a compound inequality like $1 < x < 3$ or like $x < 1$ or $x > 3$ . Use integers, proper fractions, or improper fractions in simplest form.______

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Solve absolute value inequalities

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Q. Solve for

r

.

|r - 8| \geq 8

Write a compound inequality like

1 < x < 3

or like

x < 1

or

x > 3

. Use integers, proper fractions, or improper fractions in simplest form.______

Absolute Value Inequality: We have the inequality $|r - 8| \geq 8$ . The absolute value inequality means that the expression inside the absolute value, $r - 8$ , can either be greater than or equal to $8$ or less than or equal to $-8$ .
Case $1$ : $r - 8 \geq 8$ : First, let's solve for the case where $r - 8$ is greater than or equal to $8$ : $r - 8 \geq 8$ Add $8$ to both sides to isolate $r$ : $r - 8 + 8 \geq 8 + 8$ $r \geq 16$
Case $2$ : $r - 8 \leq -8$ : Now, let's solve for the case where $r - 8$ is less than or equal to $-8$ : $r - 8 \leq -8$ Add $8$ to both sides to isolate $r$ : $r - 8 + 8 \leq -8 + 8$ $r \leq 0$
Compound Inequality Solution: Combining both cases into a compound inequality, we get: $\newline$ $r \leq 0$ or $r \geq 16$ $\newline$ This is the solution to the inequality $|r - 8| \geq 8$ .

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