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Solve for $w$ . $\newline$ |w + 8| < 9 $\newline$ 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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Solve absolute value inequalities

Full solution

Q. Solve for

w

.

\newline

|w + 8| < 9

\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

______

Understand Absolute Value Inequality: First, we need to understand the absolute value inequality |w + 8| < 9. This means that the expression inside the absolute value, $w + 8$ , must be less than $9$ units away from zero on the number line. We will split this into two separate inequalities to solve for $w$ .
Case of Positive or Zero Expression: We consider the case when the expression inside the absolute value is positive or zero. This gives us the inequality w + 8 < 9. We will solve for $w$ by subtracting $8$ from both sides of the inequality. $\newline$ w + 8 - 8 < 9 - 8 $\newline$ w < 1
Case of Negative Expression: Now we consider the case when the expression inside the absolute value is negative. This gives us the inequality - (w + 8) < 9. We will solve for $w$ by first distributing the negative sign and then adding $8$ to both sides of the inequality. $\newline$ - (w + 8) < 9 $\newline$ -w - 8 < 9 $\newline$ -w < 9 + 8 $\newline$ -w < 17 $\newline$ Multiplying both sides by $-1$ (and remembering to reverse the inequality sign because we are multiplying by a negative number) gives us: $\newline$ w > -17
Combining Inequalities: Combining the two inequalities from the previous steps, we get a compound inequality that represents all the values of $w$ that satisfy the original absolute value inequality: $\newline$ -17 < w < 1

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