Question

solve the following inequality and graph the solution set on a number line.
-7 < x + 1 < -2

Explanation:

Step1: Subtract 1 from all parts

To solve the compound inequality \(-7 < x + 1 < -2\), we subtract 1 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 \(d\).

Subtracting 1 from each part:

$$ -7 - 1 < x + 1 - 1 < -2 - 1 $$

Step2: Simplify each part

Simplify the left, middle, and right parts of the inequality:

Left part: \(-7 - 1=-8\)

Middle part: \(x + 1 - 1 = x\)

Right part: \(-2 - 1=-3\)

So the simplified inequality is:

$$ -8 < x < -3 $$

To graph this on a number line:

1. Draw a number line.
2. Locate the points \(-8\) and \(-3\) on the number line.
3. Since the inequality is strict ( \(<\) , not \(\leq\) or \(\geq\) ), we use open circles at \(-8\) and \(-3\).
4. Draw a line segment connecting the open circles at \(-8\) and \(-3\) to represent all the numbers \(x\) such that \(-8 < x < -3\).

Answer:

The solution to the inequality \(-7 < x + 1 < -2\) is \(-8 < x < -3\). The graph on the number line has open circles at \(-8\) and \(-3\) with a line segment connecting them.