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].
### 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].
