Question

solve for b and graph the solution. 20 < |b + 50| 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 the inequality \( 20<|b + 50|\). We know that the absolute - value inequality \(|x|>a\) (where \(a>0\)) is equivalent to \(x < - a\) or \(x>a\).
Here, \(x = b + 50\) and \(a = 20\). So we can split the inequality into two cases:
Case 1: \(b + 50 < - 20\)
Subtract 50 from both sides of the inequality: \(b+50 - 50<-20 - 50\), which simplifies to \(b < - 70\).
Case 2: \(b + 50>20\)
Subtract 50 from both sides of the inequality: \(b + 50-50>20 - 50\), which simplifies to \(b>-30\).

Step2: Graph the solution

• For the inequality \(b < - 70\), on the number line, we use an open circle at \(b=-70\) (because the inequality is strict, \(b

eq - 70\)) and draw an arrow to the left (since \(b\) is less than \(-70\)).

• For the inequality \(b>-30\), on the number line, we use an open circle at \(b = - 30\) (because the inequality is strict, \(b

eq-30\)) and draw an arrow to the right (since \(b\) is greater than \(-30\)).

Answer:

The solution for \( b \) is \( b < -30 \) or \( b > -70 \), and the graph has two rays: one starting at \( -30 \) (open circle) going left, and one starting at \( -70 \) (open circle) going right.