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Find solutions: (1)/(2)(x-4) <= (1)/(3)(x+1)+2

Find solutions: 12(x−4)≤13(x+1)+2\frac{1}{2}(x-4) \leq \frac{1}{3}(x+1)+2

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Full solution

Q. Find solutions: 12(x−4)≤13(x+1)+2\frac{1}{2}(x-4) \leq \frac{1}{3}(x+1)+2
  1. Clear Fractions: Clear the fractions by finding a common denominator.\newlineThe common denominator for 22 and 33 is 66. Multiply both sides of the inequality by 66 to eliminate the fractions.\newline$\(6\) \times \left(\frac{\(1\)}{\(2\)}\right)(x - \(4\)) \leq \(6\) \times \left(\left(\frac{\(1\)}{\(3\)}\right)(x + \(1\)) + \(2\)\right)
  2. Distribute \(6\): Distribute the \(6\) on both sides of the inequality.\[(\frac{6}{2})(x - 4) \leq (\frac{6}{3})(x + 1) + 6 \times 2\]
  3. Simplify Multiplication: Simplify the multiplication. \(3(x - 4) \leq 2(x + 1) + 12\)
  4. Distribute \(3\) and \(2\): Distribute the \(3\) and the \(2\) on both sides of the inequality.\(3x - 12 \leq 2x + 2 + 12\)
  5. Combine Like Terms: Combine like terms on the right side of the inequality. \(3x - 12 \leq 2x + 14\)
  6. Subtract \(2x\): Subtract \(2x\) from both sides to get the \(x\) terms on one side.\(\newline\)\(3x - 2x - 12 \leq 14\)
  7. Simplify Subtraction: Simplify the subtraction on the left side. \(x - 12 \leq 14\)
  8. Add \(12\): Add \(12\) to both sides to isolate \(x\).\(\newline\)\(x \leq 14 + 12\)
  9. Simplify Addition: Simplify the addition on the right side. \(x \leq 26\)

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