Question

solve for d and graph the solution.
300 ≤ 3|d + 900|
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.

Explanation:

Step1: Solve the absolute - value inequality

Given \(300\leq3|d + 900|\).
First, divide both sides by \(3\): \(\frac{300}{3}\leq|d + 900|\), so \(100\leq|d + 900|\).
By the definition of absolute value \(|x|=a(a\geq0)\) implies \(x = a\) or \(x=-a\), and \(|x|\geq a(a\geq0)\) implies \(x\geq a\) or \(x\leq - a\).
For \(|d + 900|\geq100\), we have two cases:
Case 1: \(d+900\geq100\). Subtract \(900\) from both sides: \(d+900-900\geq100 - 900\), so \(d\geq - 800\).
Case 2: \(d + 900\leq-100\). Subtract \(900\) from both sides: \(d+900-900\leq-100 - 900\), so \(d\leq-1000\).

Step2: Graph the solution

The solution of the inequality \(300\leq3|d + 900|\) is \(d\leq-1000\) or \(d\geq - 800\).
On the number - line:

• For \(d\leq-1000\), we draw a ray starting at \(-1000\) (with a filled - in circle since the inequality includes equality) and going to the left.
• For \(d\geq - 800\), we draw a ray starting at \(-800\) (with a filled - in circle since the inequality includes equality) and going to the right.

Answer:

The solution of the inequality \(300\leq3|d + 900|\) is \(d\leq-1000\) or \(d\geq - 800\). On the number - line, we have a ray starting at \(-1000\) (filled - in circle) going left and a ray starting at \(-800\) (filled - in circle) going right.