Question

1. solve

$-\frac{2}{3}(\frac{1}{4}x - \frac{4}{5})=\frac{5}{12}x - 1$

Explanation:

Step1: Distribute the left - hand side

\[-\frac{2}{3}\times\frac{1}{4}x+\frac{2}{3}\times\frac{4}{5}=\frac{5}{12}x - 1\]
\[-\frac{1}{6}x+\frac{8}{15}=\frac{5}{12}x - 1\]

Step2: Move terms with \(x\) to one side

Add \(\frac{1}{6}x\) to both sides:
\[\frac{8}{15}=\frac{5}{12}x+\frac{1}{6}x - 1\]
Find a common denominator for the \(x\) - terms on the right - hand side. The common denominator of 12 and 6 is 12. So \(\frac{1}{6}x=\frac{2}{12}x\).
\[\frac{8}{15}=\frac{5 + 2}{12}x-1\]
\[\frac{8}{15}=\frac{7}{12}x-1\]

Step3: Move the constant term to the left - hand side

Add 1 to both sides. Since \(1=\frac{15}{15}\), we have:
\[\frac{8}{15}+\frac{15}{15}=\frac{7}{12}x\]
\[\frac{8 + 15}{15}=\frac{7}{12}x\]
\[\frac{23}{15}=\frac{7}{12}x\]

Step4: Solve for \(x\)

Multiply both sides by \(\frac{12}{7}\):
\[x=\frac{23}{15}\times\frac{12}{7}\]
\[x=\frac{23\times12}{15\times7}\]
\[x=\frac{23\times4}{5\times7}\]
\[x = \frac{92}{35}\]

Answer:

\(x=\frac{92}{35}\)