Question

the area a of a rectangle with dimensions 4x and 5 is described with the inequality 100 ≤ a ≤ 1,000.

part a

select the compound inequality for the area written in terms of x.

○ a. 100 ≤ 10 + 8x ≤ 1,000

○ b. 100 ≤ 18x ≤ 1,000

○ c. 100 ≤ 9x ≤ 1,000

○ d. 100 ≤ 20x ≤ 1,000

part b

the solution to the compound inequality is choose... ≤ x ≤ choose...

are all of your solutions viable? choose...

Explanation:

Part A

Step1: Recall area of rectangle

The area \( A \) of a rectangle is given by the formula \( A=\text{length}\times\text{width} \). Here, the dimensions are \( 4x \) and \( 5 \), so \( A = 4x\times5 \).
\[ A = 20x \]

Step2: Substitute into inequality

We know that \( 100\leq A\leq1000 \). Substituting \( A = 20x \) into this inequality, we get \( 100\leq20x\leq1000 \).

Step1: Solve left inequality

We have the compound inequality \( 100\leq20x\leq1000 \). First, solve \( 100\leq20x \). Divide both sides by 20:
\[ \frac{100}{20}\leq x \]
\[ 5\leq x \]

Step2: Solve right inequality

Next, solve \( 20x\leq1000 \). Divide both sides by 20:
\[ x\leq\frac{1000}{20} \]
\[ x\leq50 \]

Step3: Combine solutions

Combining the two solutions, we get \( 5\leq x\leq50 \).

Answer:

D. \( 100 \leq 20x \leq 1,000 \)

Part B