Question

select the correct answer.
which of the following graphs shows the solution set for the inequality below?
|x + 2| + 7 > 10
a.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
b.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
c.
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
d.
-5 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4

Explanation:

Step1: Solve the inequality

Start with \(|x + 2|+7>10\). Subtract 7 from both sides: \(|x + 2|>10 - 7\), so \(|x + 2|>3\).

Step2: Analyze the absolute - value inequality

For \(|a|>b\) (where \(b>0\)), it is equivalent to \(a>b\) or \(a < - b\). Here, \(a=x + 2\) and \(b = 3\), so we have two inequalities:

• Inequality 1: \(x+2>3\). Subtract 2 from both sides: \(x>3 - 2\), so \(x>1\).
• Inequality 2: \(x + 2<-3\). Subtract 2 from both sides: \(x<-3 - 2\), so \(x<-5\).

Step3: Match with the number - line graphs

The solution set \(x>1\) or \(x < - 5\) means that on the number line, we have an open circle at \(x=-5\) (since \(x=-5\) is not included) and the line is shaded to the left of \(-5\), and an open circle at \(x = 1\) (since \(x = 1\) is not included) and the line is shaded to the right of \(1\). Looking at the options, option C has an open circle at \(-5\) with shading to the left and an open circle at \(1\) with shading to the right.

Answer:

C