To solve the compound inequality [tex]\(-5 < 2x + 3 \leq 22\)[/tex], we'll break it down into two separate inequalities and solve each one step-by-step. Let's start with the first inequality.
### Step 1: Solve [tex]\(-5 < 2x + 3\)[/tex]
Subtract 3 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
-5 - 3 < 2x
\][/tex]
[tex]\[
-8 < 2x
\][/tex]
Now, divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{-8}{2} < x
\][/tex]
[tex]\[
-4 < x
\][/tex]
This simplifies to:
[tex]\[
x > -4
\][/tex]
### Step 2: Solve [tex]\(2x + 3 \leq 22\)[/tex]
Subtract 3 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
2x + 3 - 3 \leq 22 - 3
\][/tex]
[tex]\[
2x \leq 19
\][/tex]
Now, divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[
x \leq \frac{19}{2}
\][/tex]
[tex]\[
x \leq 9.5
\][/tex]
### Step 3: Combine the results
We now have two results:
[tex]\[
x > -4
\][/tex]
[tex]\[
x \leq 9.5
\][/tex]
Combining these results into a single compound inequality, we get:
[tex]\[
-4 < x \leq 9.5
\][/tex]
### Step 4: Write the solution in interval notation
The solution to the compound inequality in interval notation is:
[tex]\[
(-4, 9.5]
\][/tex]
This means that [tex]\(x\)[/tex] is greater than [tex]\(-4\)[/tex] but less than or equal to [tex]\(9.5\)[/tex].
