Question

write and solve an inequality to find the values of x for which the perimeter of the rectangle is less than 104. select the correct answer below and fill in the answer box to complete your choice. a. x > \square b. x ≤ \boxed{104} c. x < \square d. x ≥ \square

Explanation:

Step1: Recall perimeter formula for rectangle

The perimeter \( P \) of a rectangle with length \( l \) and width \( w \) is given by \( P = 2(l + w) \). Here, the length is \( x + 4 \) and the width is \( x \). So the perimeter \( P = 2(x + (x + 4)) \).

Step2: Simplify the perimeter expression

Simplify \( 2(x + x + 4) = 2(2x + 4) = 4x + 8 \).

Step3: Set up the inequality

We want the perimeter to be less than 104, so we set up the inequality \( 4x + 8 < 104 \).

Step4: Solve the inequality

Subtract 8 from both sides: \( 4x + 8 - 8 < 104 - 8 \), which simplifies to \( 4x < 96 \).
Then divide both sides by 4: \( \frac{4x}{4} < \frac{96}{4} \), so \( x < 24 \). Wait, but looking at the options, maybe there was a misinterpretation. Wait, maybe the perimeter is less than or equal? Wait no, the problem says "less than 104". Wait, maybe the original problem's rectangle has length \( x \) and width \( x + 4 \), but let's recheck. Wait, maybe the perimeter formula was misapplied. Wait, no, perimeter of rectangle is \( 2(l + w) \). If length is \( x \) and width is \( x + 4 \), then perimeter is \( 2(x + x + 4) = 4x + 8 \). We need \( 4x + 8 < 104 \). Solving: \( 4x < 96 \), \( x < 24 \). But the options given are A. \( x > \) [blank], B. \( x \leq 104 \), C. \( x < \) [blank], D. \( x \geq \) [blank]. Wait, maybe the problem was about the perimeter being less than 104, so the inequality is \( 2(x + (x + 4)) < 104 \), which simplifies to \( 2(2x + 4) < 104 \), then \( 4x + 8 < 104 \), \( 4x < 96 \), \( x < 24 \). So the correct inequality is \( x < 24 \), so option C with the blank filled as 24. Wait, but maybe the original problem had a different setup. Wait, maybe the length is \( x \) and the other side is \( x + 4 \), and perimeter is less than 104. So solving \( 2(x + x + 4) < 104 \), \( 2(2x + 4) < 104 \), \( 4x + 8 < 104 \), \( 4x < 96 \), \( x < 24 \). So the answer for the inequality is \( x < 24 \), so option C with 24 in the box.

Answer:

C. \( x < 24 \)