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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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