Question

solve the following inequality and graph the solution set on a number line.
4 < x + 9 < 6

Explanation:

Step1: Subtract 9 from all parts

To solve the compound inequality \(4 < x + 9 < 6\), we subtract 9 from each part of the inequality. This is based on the property of inequalities that if \(a < b < c\), then \(a - d < b - d < c - d\) for any real number \(d\). So we have:
\(4 - 9 < x + 9 - 9 < 6 - 9\)

Step2: Simplify each part

Simplifying the left - hand side: \(4-9=-5\)
Simplifying the middle part: \(x + 9-9=x\)
Simplifying the right - hand side: \(6 - 9=-3\)
So the solution to the inequality \(4 < x + 9 < 6\) is \(-5 < x < - 3\)

For graphing on the number line:

• We draw a number line.
• We mark the points \(-5\) and \(-3\) with open circles (since the inequality is strict, \(x\) is not equal to \(-5\) or \(-3\)).
• Then we shade the region between \(-5\) and \(-3\) on the number line.

Answer:

The solution to the inequality \(4 < x + 9 < 6\) is \(-5 < x < - 3\). The graph of the solution set on the number line has open circles at \(-5\) and \(-3\) and the region between them is shaded.