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