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Solve for $b$ . $|b| - 7 \leq 7$ 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.______

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Solve absolute value inequalities

Full solution

Q. Solve for

b

.

|b| - 7 \leq 7

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.______

Isolate absolute value: First, we need to isolate the absolute value expression on one side of the inequality. $\newline$ $|b| - 7 \leq 7$ $\newline$ Add $7$ to both sides to isolate the absolute value. $\newline$ $|b| - 7 + 7 \leq 7 + 7$ $\newline$ $|b| \leq 14$
Consider absolute value definition: Now we need to consider the definition of absolute value, which states that |b|\(\newline) is the distance of b\(\newline) from \(0 $\newline$ ) on the number line. This means that b\(\newline) can be either positive or negative, but not farther than \(14 $\newline$ ) units away from \(0 $\newline$ ). $\newline$ So we have two cases: $\newline$ Case $1$ : b\(\newline) is non-negative, which gives us b \leq \(14 $\newline$ ). $\newline$ Case $2$ : b\(\newline) is negative, which gives us -b \leq \(14 $\newline$ ), or equivalently, b \geq \(-14 $\newline$ ) after multiplying both sides by \(-1 $\newline$ ) and reversing the inequality sign.
Combine cases into compound inequality: Combining both cases into a compound inequality, we get: $\newline$ $-14 \leq b \leq 14$ $\newline$ This compound inequality represents all the values of $b$ that satisfy the original inequality $|b| - 7 \leq 7$ .

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