Question

solve for x.

$2\frac{1}{2}x - \frac{3}{4}(2x + 5) = \frac{3}{8}$

$\circ \\ x = -4\frac{5}{8}$

$\circ \\ x = -3\frac{3}{8}$

$\circ \\ x = 4\frac{1}{8}$

$\circ \\ x = 5\frac{3}{8}$

Explanation:

Step1: Convert mixed number to improper fraction

The mixed number \(2\frac{1}{2}\) can be converted to an improper fraction. \(2\frac{1}{2}=\frac{2\times2 + 1}{2}=\frac{5}{2}\). So the equation becomes \(\frac{5}{2}x-\frac{3}{4}(2x + 5)=\frac{3}{8}\).

Step2: Distribute the \(-\frac{3}{4}\)

Using the distributive property \(a(b + c)=ab+ac\), we have \(\frac{5}{2}x-\frac{3}{4}\times2x-\frac{3}{4}\times5=\frac{3}{8}\). Simplify the terms: \(\frac{5}{2}x-\frac{3}{2}x-\frac{15}{4}=\frac{3}{8}\).

Step3: Combine like terms for \(x\)

Subtract the \(x\) terms: \(\frac{5}{2}x-\frac{3}{2}x=\frac{5 - 3}{2}x=\frac{2}{2}x=x\). So the equation is now \(x-\frac{15}{4}=\frac{3}{8}\).

Step4: Solve for \(x\)

Add \(\frac{15}{4}\) to both sides of the equation. First, find a common denominator for \(\frac{15}{4}\) and \(\frac{3}{8}\), which is 8. \(\frac{15}{4}=\frac{15\times2}{4\times2}=\frac{30}{8}\). Then \(x=\frac{3}{8}+\frac{30}{8}=\frac{3 + 30}{8}=\frac{33}{8}\). Convert \(\frac{33}{8}\) to a mixed number: \(33\div8 = 4\) with a remainder of 1, so \(\frac{33}{8}=4\frac{1}{8}\).

Answer:

\(x = 4\frac{1}{8}\) (corresponding to the option \(x = 4\frac{1}{8}\))