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Solve for $c$ . $|c| - 7 < -5$ 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

c

.

|c| - 7 < -5

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: We have the inequality: $\newline$ $|c| - 7 < -5$ $\newline$ First, we need to isolate the absolute value term $|c|$ on one side of the inequality. $\newline$ $|c| - 7 + 7 < -5 + 7$ $\newline$ $|c| < 2$
Consider absolute value definition: Now we need to consider the definition of absolute value. The inequality $|c| < 2$ means that $c$ is less than $2$ units away from $0$ on the number line. This gives us two inequalities: $\newline$ $c < 2$ and $-c < 2$
Multiply by $-1$ : The second inequality, $-c < 2$ , can be multiplied by $-1$ to get $c > -2$ . Remember that multiplying or dividing an inequality by a negative number reverses the inequality sign. $\newline$ $-c < 2$ $\newline$ $(-1)(-c) > (-1)(2)$ $\newline$ $c > -2$
Combine inequalities: Combining the two inequalities from the previous steps, we get the compound inequality: $\newline$ $-2 < c < 2$ $\newline$ This is the solution to the original inequality $|c| - 7 < -5$ .

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