Question

1. \\(\frac{5 + 2m}{6} = 4\\)\
2. \\(\frac{7}{8}d - 4 = 10\\)\
3. \\(|2x - 4| = 10\\)

Explanation:

Problem 21: Solve \(\boldsymbol{\frac{5 + 2m}{6}=4}\)

Step1: Multiply both sides by 6

To eliminate the denominator, multiply both sides of the equation by 6:
\(6\times\frac{5 + 2m}{6}=4\times6\)
Simplifies to: \(5 + 2m = 24\)

Step2: Subtract 5 from both sides

Subtract 5 from each side to isolate the term with \(m\):
\(5 + 2m - 5 = 24 - 5\)
Simplifies to: \(2m = 19\)

Step3: Divide by 2

Divide both sides by 2 to solve for \(m\):
\(\frac{2m}{2}=\frac{19}{2}\)
Simplifies to: \(m=\frac{19}{2}=9.5\)

Step1: Add 4 to both sides

Add 4 to each side to isolate the term with \(d\):
\(\frac{7}{8}d - 4 + 4 = 10 + 4\)
Simplifies to: \(\frac{7}{8}d = 14\)

Step2: Multiply by \(\frac{8}{7}\)

Multiply both sides by \(\frac{8}{7}\) to solve for \(d\):
\(\frac{8}{7}\times\frac{7}{8}d = 14\times\frac{8}{7}\)
Simplifies to: \(d = 16\)

Step1: Apply absolute value definition

The absolute value equation \(|A| = B\) (where \(B\geq0\)) implies \(A = B\) or \(A = -B\). So:
\(2x - 4 = 10\) or \(2x - 4 = -10\)

Step2: Solve \(2x - 4 = 10\)

Add 4 to both sides: \(2x - 4 + 4 = 10 + 4\) → \(2x = 14\)
Divide by 2: \(\frac{2x}{2}=\frac{14}{2}\) → \(x = 7\)

Step3: Solve \(2x - 4 = -10\)

Add 4 to both sides: \(2x - 4 + 4 = -10 + 4\) → \(2x = -6\)
Divide by 2: \(\frac{2x}{2}=\frac{-6}{2}\) → \(x = -3\)

Answer:

\(m = \frac{19}{2}\) (or \(m = 9.5\))

Problem 24: Solve \(\boldsymbol{\frac{7}{8}d - 4 = 10}\)