Solve for r. |r + 8| ≥ 2 Write a compound inequality like 1 3. Use integers, proper fractions, or improper fractions in simplest form. ______

Absolute Value Inequality: We have the inequality |r + 8| ≥ 2. The absolute value inequality means that the expression inside the absolute value, r + 8, can either be greater than or equal to 2 or less than or equal to -2. We will split this into two separate inequalities to solve for r.Case 1: Non-Negative Expression: First, let's consider the case where the expression inside the absolute value is non-negative: r + 8 ≥ 2. We will solve for r by subtracting 8 from both sides of the inequality. r + 8 - 8 ≥ 2 - 8 r ≥ -6Case 2: Negative Expression: Now, let's consider the case where the expression inside the absolute value is negative: r + 8 ≤ -2. We will solve for r by subtracting 8 from both sides of the inequality. r + 8 - 8 ≤ -2 - 8 r ≤ -10Combining Cases: Combining the two cases, we have a compound inequality: r ≤ -10 or r ≥ -6. This is the solution to the original absolute value inequality.

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Solve for $r$ . $|r + 8| \geq 2$ 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. $\newline$ ______

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Algebra 2

Solve absolute value inequalities

Full solution

Q. Solve for

r

|r + 8| \geq 2

Write a compound inequality like

1 < x < 3

or like

x < 1

x > 3

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

\newline

______

Absolute Value Inequality: We have the inequality $|r + 8| \geq 2$ . The absolute value inequality means that the expression inside the absolute value, $r + 8$ , can either be greater than or equal to $2$ or less than or equal to $-2$ . We will split this into two separate inequalities to solve for $r$ .
Case $1$ : Non-Negative Expression: First, let's consider the case where the expression inside the absolute value is non-negative: $r + 8 \geq 2$ . We will solve for $r$ by subtracting $8$ from both sides of the inequality. $\newline$ $r + 8 - 8 \geq 2 - 8$ $\newline$ $r \geq -6$
Case $2$ : Negative Expression: Now, let's consider the case where the expression inside the absolute value is negative: $r + 8 \leq -2$ . We will solve for $r$ by subtracting $8$ from both sides of the inequality. $\newline$ $r + 8 - 8 \leq -2 - 8$ $\newline$ $r \leq -10$
Combining Cases: Combining the two cases, we have a compound inequality: $r \leq -10$ or $r \geq -6$ . This is the solution to the original absolute value inequality.