Math Problem Statement

Solve the inequality

Solve the inequality (3x - 1) / -4 < 8 - (x + 3) / -6.

Solution

The inequality in the image is:

3x14<8x+36\frac{3x - 1}{-4} < 8 - \frac{x + 3}{-6}

Step-by-step solution:

Step 1: Simplify both sides

First, simplify the right-hand side: x+36=x+36=x636=x612\frac{x + 3}{-6} = -\frac{x + 3}{6} = -\frac{x}{6} - \frac{3}{6} = -\frac{x}{6} - \frac{1}{2} So, the inequality becomes: 3x14<8+x6+12\frac{3x - 1}{-4} < 8 + \frac{x}{6} + \frac{1}{2}

Step 2: Eliminate fractions by finding a common denominator

The least common denominator between 4-4, 6, and 2 is 12. Multiply every term by 12 to eliminate the fractions: 12×(3x14)<12×(8+x6+12)12 \times \left(\frac{3x - 1}{-4}\right) < 12 \times \left(8 + \frac{x}{6} + \frac{1}{2}\right) This simplifies to: 3×(3x1)<12×8+2x+63 \times (3x - 1) < 12 \times 8 + 2x + 6

Simplifying both sides: 9x3<96+2x+69x - 3 < 96 + 2x + 6 9x3<102+2x9x - 3 < 102 + 2x

Step 3: Isolate the variable

Move all terms involving xx to one side and constants to the other: 9x2x<102+39x - 2x < 102 + 3 7x<1057x < 105

Step 4: Solve for xx

Divide both sides by 7: x<15x < 15

Final Solution:

The solution to the inequality is: x<15x < 15

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

  1. How would the inequality change if the signs of the fractions were reversed?
  2. What happens if the coefficient of xx on one side is negative?
  3. Can you graph the solution x<15x < 15 on a number line?
  4. How would the solution change if the inequality was \leq instead of <<?
  5. What happens if there are no fractions involved?

Tip: Always be careful when multiplying or dividing an inequality by a negative number, as this will reverse the inequality sign.

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Math Problem Analysis

Mathematical Concepts

Algebra
Inequalities
Linear Equations

Formulas

Fraction simplification
Inequality solving
Variable isolation

Theorems

Properties of inequalities
Multiplication of inequalities by negative numbers

Suitable Grade Level

Grades 9-11