Question

solve the following inequality. graph the solution set. |x - 5| + 5 < 7
a. there are infinitely many solutions. the solution set is (type your answer in interval notation.)
b. there are finitely many solutions. the solution set is (use a comma to separate answers as needed.)
c. there is no real solution.
graph the solution set. choose the correct graph below.

Explanation:

Step1: Simplify the inequality

Subtract 5 from both sides of \(|x - 5|+5\lt7\).
We get \(|x - 5|\lt2\).

Step2: Solve the absolute - value inequality

By the property of absolute - value \(|a|\lt b\) (\(b\gt0\)) which is equivalent to \(-b\lt a\lt b\).
Here \(a=x - 5\) and \(b = 2\), so \(-2\lt x-5\lt2\).

Step3: Isolate \(x\)

Add 5 to all parts of the compound inequality \(-2\lt x - 5\lt2\).
For the left - hand side: \(-2+5\lt x-5 + 5\), which gives \(3\lt x\).
For the right - hand side: \(x-5 + 5\lt2+5\), which gives \(x\lt7\).

In interval notation, the solution set is \((3,7)\).
When graphing the solution set \((3,7)\) on a number line, we have an open circle at \(x = 3\) and an open circle at \(x = 7\) and the line segment between them.

Answer:

A. There are infinitely many solutions. The solution set is \((3,7)\) (Type your answer in interval notation). And the correct graph is C (assuming graph C has open - circles at 3 and 7 and a line segment between them).