Question

select the correct answer. which of the following graphs shows the solution set for the inequality below? |x + 2| + 7 > 10 a. graph a b. graph b c. graph c d. graph d

Explanation:

Step1: Solve the inequality \(|x + 2|+7>10\)

Subtract 7 from both sides: \(|x + 2|>10 - 7\)
Simplify: \(|x + 2|>3\)

Step2: Solve the absolute - value inequality

The absolute - value inequality \(|a|>b\) (where \(b>0\)) is equivalent to \(a>b\) or \(a < - b\).
For \(|x + 2|>3\), we have two cases:
Case 1: \(x+2>3\)
Subtract 2 from both sides: \(x>3 - 2\), so \(x > 1\)
Case 2: \(x + 2<-3\)
Subtract 2 from both sides: \(x<-3 - 2\), so \(x<-5\)

Step3: Analyze 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 \(x
eq - 5\)) and an arrow to the left of \(-5\), and an open circle at \(x = 1\) (since \(x
eq1\)) and an arrow to the right of \(1\).
Looking at the options, option C has an open circle at \(x = 1\) (arrow to the right) and an open circle at \(x=-5\) (arrow to the left), which matches our solution.

Answer:

C