Solve for r. |r + 2| ≤ 7 Write a compound inequality like 1 3. Use integers, proper fractions, or improper fractions in simplest form. ______

Absolute Value Definition: We have the inequality |r + 2| ≤ 7. To solve for r, we need to consider the definition of absolute value, which states that |x| ≤ a implies -a ≤ x ≤ a. We will apply this definition to the inequality at hand.Applying Definition: Applying the definition of absolute value to our inequality, we get -7 ≤ r + 2 ≤ 7. This gives us two separate inequalities to solve: r + 2 ≤ 7 and r + 2 ≥ -7.Solving r + 2 ≤ 7: First, we solve the inequality r + 2 ≤ 7. Subtract 2 from both sides to isolate r: r ≤ 7 - 2, which simplifies to r ≤ 5.Solving r + 2 ≥ -7: Next, we solve the inequality r + 2 ≥ -7. Subtract 2 from both sides to isolate r: r ≥ -7 - 2, which simplifies to r ≥ -9.Combining Inequalities: Combining the two inequalities, we get the compound inequality -9 ≤ r ≤ 5. This is the solution to the original inequality |r + 2| ≤ 7.

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Solve for $r$ . $\newline$ $\lvert r + 2 \rvert \leq 7$ $\newline$ 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

\newline

\lvert r + 2 \rvert \leq 7

\newline

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 Definition: We have the inequality $|r + 2| \leq 7$ . To solve for $r$ , we need to consider the definition of absolute value, which states that $|x| \leq a$ implies $-a \leq x \leq a$ . We will apply this definition to the inequality at hand.
Applying Definition: Applying the definition of absolute value to our inequality, we get $-7 \leq r + 2 \leq 7$ . This gives us two separate inequalities to solve: $r + 2 \leq 7$ and $r + 2 \geq -7$ .
Solving $r + 2 \leq 7$ : First, we solve the inequality $r + 2 \leq 7$ . Subtract $2$ from both sides to isolate $r$ : $r \leq 7 - 2$ , which simplifies to $r \leq 5$ .
Solving $r + 2 \geq -7$ : Next, we solve the inequality $r + 2 \geq -7$ . Subtract $2$ from both sides to isolate $r$ : $r \geq -7 - 2$ , which simplifies to $r \geq -9$ .
Combining Inequalities: Combining the two inequalities, we get the compound inequality $-9 \leq r \leq 5$ . This is the solution to the original inequality $|r + 2| \leq 7$ .