Solve the compound inequality:

[tex]-5 \ \textless \ 2x + 3 \leq 22[/tex]

Enter the exact answer in interval notation.

To enter [tex]\infty[/tex], type infinity. To enter [tex]U[/tex], type U.



Answer :

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].