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Solve for $d$ .|d| - 1 < 17Write 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

d

.

|d| - 1 < 17

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: $\newline$ |d| - 1 < 17 $\newline$ First, we isolate the absolute value term by adding $1$ to both sides of the inequality. $\newline$ |d| - 1 + 1 < 17 + 1 $\newline$ |d| < 18
Consider Absolute Value Definition: Now we need to consider the definition of absolute value. The inequality |d| < 18 means that $d$ is less than $18$ units away from $0$ on the number line. This gives us two inequalities: $\newline$ d < 18 and -d < 18
Multiply by $-1$ : The second inequality, -d < 18, can be multiplied by $-1$ to get d > -18. Remember that multiplying or dividing an inequality by a negative number reverses the inequality sign. $\newline$ -d < 18 $\newline$ (-1)(-d) > (-1)(18) $\newline$ d > -18
Combine Inequalities: Combining the two inequalities from the previous steps, we get the compound inequality: $\newline$ -18 < d < 18 $\newline$ This means that $d$ is greater than $-18$ and less than $18$ .

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