To solve the equation [tex]\(1+\frac{3}{4 x}=\frac{2}{5}\)[/tex], let's follow these steps to find the solution set:
1. Isolate the fraction:
We need to get the fraction [tex]\(\frac{3}{4x}\)[/tex] by itself. Start by subtracting 1 from both sides of the equation:
[tex]\[ 1 + \frac{3}{4x} - 1 = \frac{2}{5} - 1 \][/tex]
[tex]\[ \frac{3}{4x} = \frac{2}{5} - 1 \][/tex]
2. Simplify the right side:
Find a common denominator for the right side to subtract the fractions:
[tex]\[ \frac{2}{5} - 1 = \frac{2}{5} - \frac{5}{5} = \frac{2 - 5}{5} = \frac{-3}{5} \][/tex]
This simplifies to:
[tex]\[ \frac{3}{4x} = \frac{-3}{5} \][/tex]
3. Solve for [tex]\(x\)[/tex]:
Cross-multiply to solve for [tex]\(x\)[/tex]:
[tex]\[ 3 \cdot 5 = -3 \cdot 4x \][/tex]
[tex]\[ 15 = -12x \][/tex]
Divide both sides by -12 to isolate [tex]\(x\)[/tex]:
[tex]\[ x = \frac{15}{-12} = -\frac{15}{12} \][/tex]
Simplify the fraction:
[tex]\[ x = -\frac{5}{4} \][/tex]
4. Identify the correct answer:
Among the given choices:
A. [tex]\(\left\{ -\frac{5}{4} \right\} \)[/tex]
B. [tex]\(\left\{ \frac{4}{5} \right\} \)[/tex]
C. [tex]\(\left\{ -\frac{4}{5} \right\} \)[/tex]
D. [tex]\(\left\{ \frac{5}{4} \right\} \)[/tex]
The solution we've found is:
[tex]\( x = -\frac{5}{4} \)[/tex]
Therefore, the correct answer is:
A. [tex]\(\left\{-\frac{5}{4}\right\}\)[/tex]
