Question

1. solve the inequality $-5 < x + 2 < 3$. graph the solution on the number line.

Explanation:

Step1: Subtract 2 from all parts

To solve the compound inequality \(-5 < x + 2 < 3\), we subtract 2 from each part of the inequality. This is based on the property of inequalities that if we perform the same operation on all parts of an inequality, the inequality remains valid. So we have:
\(-5 - 2 < x + 2 - 2 < 3 - 2\)

Step2: Simplify each part

Simplifying the left - hand side: \(-5-2=-7\)
Simplifying the middle part: \(x + 2-2=x\)
Simplifying the right - hand side: \(3 - 2 = 1\)
So the solution to the inequality is \(-7

To graph this on the number line:

• We draw an open circle at \(x=-7\) (because \(x\) is not equal to \(-7\)) and an open circle at \(x = 1\) (because \(x\) is not equal to \(1\)).
• Then we draw a line segment connecting the two open circles to represent all the values of \(x\) that satisfy the inequality.

Answer:

The solution to the inequality \(-5 < x + 2 < 3\) is \(-7