Solve for t. |t + 8| ≤ 4 Write a compound inequality like 1 3. Use integers, proper fractions, or improper fractions in simplest form. ______

Absolute Value Inequality: We have the inequality |t + 8| ≤ 4. The absolute value inequality means that the expression inside the absolute value, t + 8, can be either less than or equal to 4 or greater than or equal to -4. We will split this into two separate inequalities to solve for t.Case 1: Non-Negative Expression: First, let's consider the case where the expression inside the absolute value is non-negative: t + 8 ≤ 4. To find the value of t, we subtract 8 from both sides of the inequality. t + 8 - 8 ≤ 4 - 8 t ≤ -4Case 2: Non-Positive Expression: Now, let's consider the case where the expression inside the absolute value is non-positive: -(t + 8) ≤ 4. This is equivalent to -t - 8 ≤ 4. To find the value of t, we first add 8 to both sides of the inequality. -t - 8 + 8 ≤ 4 + 8 -t ≤ 12 Multiplying both sides by -1 (and remembering to reverse the inequality sign because we are multiplying by a negative number), we get: t ≥ -12Combining Inequalities: Combining the two inequalities we have found, we get the compound inequality: -12 ≤ t ≤ -4 This means that t must be greater than or equal to -12 and less than or equal to -4.

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Solve for $t$ . $\newline$ $|t + 8| \leq 4$ $\newline$ 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 1

Solve absolute value inequalities

Full solution

Q. Solve for

t

\newline

|t + 8| \leq 4

\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 Inequality: We have the inequality $|t + 8| \leq 4$ . The absolute value inequality means that the expression inside the absolute value, $t + 8$ , can be either less than or equal to $4$ or greater than or equal to $-4$ . We will split this into two separate inequalities to solve for $t$ .
Case $1$ : Non-Negative Expression: First, let's consider the case where the expression inside the absolute value is non-negative: $t + 8 \leq 4$ . To find the value of $t$ , we subtract $8$ from both sides of the inequality. $\newline$ $t + 8 - 8 \leq 4 - 8$ $\newline$ $t \leq -4$
Case $2$ : Non-Positive Expression: Now, let's consider the case where the expression inside the absolute value is non-positive: $- (t + 8) \leq 4$ . This is equivalent to $-t - 8 \leq 4$ . To find the value of $t$ , we first add $8$ to both sides of the inequality. $-t - 8 + 8 \leq 4 + 8$ $-t \leq 12$ Multiplying both sides by $-1$ (and remembering to reverse the inequality sign because we are multiplying by a negative number), we get: $t \geq -12$
Combining Inequalities: Combining the two inequalities we have found, we get the compound inequality: $\newline$ $-12 \leq t \leq -4$ $\newline$ This means that $t$ must be greater than or equal to $-12$ and less than or equal to $-4$ .