Answer :
To solve the inequality [tex]\(5 < 2x + 3 \leq 9\)[/tex] and find how many integer values [tex]\(x\)[/tex] can take, we’ll break it down into clear steps.
Step 1: Simplify the inequality by isolating the term involving [tex]\(x\)[/tex].
[tex]\[5 < 2x + 3 \leq 9\][/tex]
Step 2: Subtract 3 from all parts of the inequality.
[tex]\[ 5 - 3 < 2x + 3 - 3 \leq 9 - 3 \][/tex]
This simplifies to:
[tex]\[ 2 < 2x \leq 6 \][/tex]
Step 3: Divide all parts of the inequality by 2 to solve for [tex]\(x\)[/tex].
[tex]\[ \frac{2}{2} < \frac{2x}{2} \leq \frac{6}{2} \][/tex]
This results in:
[tex]\[ 1 < x \leq 3 \][/tex]
Step 4: Determine the integer values that [tex]\(x\)[/tex] can take. Since [tex]\(x\)[/tex] must be greater than 1 and less than or equal to 3, the possible integer values for [tex]\(x\)[/tex] are 2 and 3.
Step 5: Count the integer values identified in the previous step. The integers 2 and 3 are the solutions.
Therefore, there are [tex]\(\boxed{2}\)[/tex] integer values that [tex]\(x\)[/tex] can take to satisfy the inequality [tex]\(5 < 2x + 3 \leq 9\)[/tex].
Step 1: Simplify the inequality by isolating the term involving [tex]\(x\)[/tex].
[tex]\[5 < 2x + 3 \leq 9\][/tex]
Step 2: Subtract 3 from all parts of the inequality.
[tex]\[ 5 - 3 < 2x + 3 - 3 \leq 9 - 3 \][/tex]
This simplifies to:
[tex]\[ 2 < 2x \leq 6 \][/tex]
Step 3: Divide all parts of the inequality by 2 to solve for [tex]\(x\)[/tex].
[tex]\[ \frac{2}{2} < \frac{2x}{2} \leq \frac{6}{2} \][/tex]
This results in:
[tex]\[ 1 < x \leq 3 \][/tex]
Step 4: Determine the integer values that [tex]\(x\)[/tex] can take. Since [tex]\(x\)[/tex] must be greater than 1 and less than or equal to 3, the possible integer values for [tex]\(x\)[/tex] are 2 and 3.
Step 5: Count the integer values identified in the previous step. The integers 2 and 3 are the solutions.
Therefore, there are [tex]\(\boxed{2}\)[/tex] integer values that [tex]\(x\)[/tex] can take to satisfy the inequality [tex]\(5 < 2x + 3 \leq 9\)[/tex].
