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Solve for $q$ . $\newline$ $|q| - 3 \leq -1$ $\newline$ 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

q

.

\newline

|q| - 3 \leq -1

\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

______

Inequality Simplification: We have the inequality: $\newline$ $|q| - 3 \leq -1$ $\newline$ First, we isolate the absolute value term by adding $3$ to both sides of the inequality. $\newline$ $|q| - 3 + 3 \leq -1 + 3$ $\newline$ $|q| \leq 2$
Absolute Value Cases: Now we need to consider the two cases for the absolute value: $\newline$ Case $1$ : $q$ is non-negative, so $|q| = q$ . $\newline$ Case $2$ : $q$ is negative, so $|q| = -q$ . $\newline$ For Case $1$ , we have: $\newline$ $q \leq 2$ $\newline$ For Case $2$ , we have: $\newline$ $-q \leq 2$ , which can be rewritten as $q \geq -2$ after multiplying both sides by $-1$ and reversing the inequality sign.
Compound Inequality Solution: Combining both cases into a compound inequality, we get: $\newline$ $-2 \leq q \leq 2$ $\newline$ This is the solution to the inequality $|q| - 3 \leq -1|$ .

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\newline

Which of the following best describes the solutions to the inequality shown?

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Question

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p

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\newline

1

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Question

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2

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\newline

1

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\newline

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\newline

1

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\newline

1

\newline

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\newline

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\newline

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\newline

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\newline

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\newline

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