Question

solve for d and graph the solution.
3 > |d + 8|
click two endpoints to graph a line segment, an endpoint and an arrowhead to graph a ray, or two arrowheads to graph a line. to change endpoints from filled - in circles to empty circles, click on them.
(number line from - 12 to 0 with ticks at - 12, - 11, - 10, - 9, - 8, - 7, - 6, - 5, - 4, - 3, - 2, - 1, 0)

Explanation:

Step1: Recall absolute value inequality rule

For \( |x| < a \) (where \( a>0 \)), the solution is \( -a < x < a \). Here, we have \( 3 > |d + 8| \), which is equivalent to \( |d + 8| < 3 \).

Step2: Apply the rule to the inequality

Using the rule, we get \( -3 < d + 8 < 3 \).

Step3: Solve the left - hand inequality

Subtract 8 from both sides of \( -3 < d + 8 \): \( -3-8 < d+8 - 8\), which simplifies to \( -11 < d \).

Step4: Solve the right - hand inequality

Subtract 8 from both sides of \( d + 8 < 3 \): \( d+8 - 8<3 - 8\), which simplifies to \( d < - 5\).

Step5: Combine the two inequalities

Combining \( -11 < d \) and \( d < -5 \), we get the solution \( -11 < d < -5 \).

To graph this solution: On the number line, we use open circles at \( d=-11 \) and \( d = - 5\) (because the inequality is strict, \( < \) not \( \leq \)) and draw a line segment between them to represent all the values of \( d \) that satisfy the inequality.

Answer:

\( -11 < d < -5 \)