Sure, let's solve the compound inequality [tex]\( -5 < 2x + 3 < 5 \)[/tex] step by step.
1. Step 1: Isolate the term with [tex]\( x \)[/tex]
- First, subtract 3 from all parts of the inequality:
[tex]\[
-5 - 3 < 2x + 3 - 3 < 5 - 3
\][/tex]
Simplifying the values, we get:
[tex]\[
-8 < 2x < 2
\][/tex]
2. Step 2: Solve for [tex]\( x \)[/tex]
- Next, divide all parts of the inequality by 2:
[tex]\[
\frac{-8}{2} < \frac{2x}{2} < \frac{2}{2}
\][/tex]
Simplifying the divisions, we get:
[tex]\[
-4 < x < 1
\][/tex]
The solution to the inequality is:
[tex]\[
-4 < x < 1
\][/tex]
3. Graphing the solution on the number line
- Draw a number line.
- Mark the points [tex]\( -4 \)[/tex] and [tex]\( 1 \)[/tex] on the number line.
- Use an open circle at [tex]\( -4 \)[/tex] to indicate that [tex]\( -4 \)[/tex] is not included in the solution.
- Use an open circle at [tex]\( 1 \)[/tex] to indicate that [tex]\( 1 \)[/tex] is not included in the solution.
- Shade the region between [tex]\( -4 \)[/tex] and [tex]\( 1 \)[/tex] to indicate that all numbers between [tex]\( -4 \)[/tex] and [tex]\( 1 \)[/tex] are included in the solution set.
The graphical representation should look like this:
```
---(--o====================o--)
-4 1
```
The open circles indicate that the endpoints [tex]\( -4 \)[/tex] and [tex]\( 1 \)[/tex] are not included in the solution. The shaded part of the number line between [tex]\( -4 \)[/tex] and [tex]\( 1 \)[/tex] represents all values of [tex]\( x \)[/tex] that satisfy the inequality [tex]\( -4 < x < 1 \)[/tex].
