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Question

Difficulty: EasySolving Linear Inequalities

For a real number xx, 12\frac{1}{2} minus 23\frac{2}{3} of xx is greater than 56\frac{5}{6}. Which of the following is the complete set of solutions for xx?

  1. A
    x>−12x > -\frac{1}{2}
  2. B
    x>2x > 2
  3. x<−12x < -\frac{1}{2}Answer
  4. D
    x<−32x < -\frac{3}{2}
  5. E
    x<−5x < -5

Answer

x<−12x < -\frac{1}{2}
Subtracting 12\frac{1}{2} from both sides of the inequality 12−23x>56\frac{1}{2} - \frac{2}{3}x > \frac{5}{6} gives −23x>26-\frac{2}{3}x > \frac{2}{6}, which simplifies to −23x>13-\frac{2}{3}x > \frac{1}{3}. Multiplying both sides by the negative fraction −32-\frac{3}{2} isolates xx on the left and reverses the inequality sign from greater than (>>) to less than (<<). Performing the multiplication on the right yields 13⋅(−32)=−12\frac{1}{3} \cdot \left(-\frac{3}{2}\right) = -\frac{1}{2}. Thus, the solution set is x<−12x < -\frac{1}{2}.

Step-by-Step Solution

1
Translate the verbal description into an algebraic inequality.
12−23x>56\frac{1}{2} - \frac{2}{3}x > \frac{5}{6}
The phrase '12\frac{1}{2} minus 23\frac{2}{3} of xx' represents the expression 12−23x\frac{1}{2} - \frac{2}{3}x, and 'is greater than' translates to the inequality symbol >>.
2
Subtract 12\frac{1}{2} from both sides of the inequality.
−23x>13-\frac{2}{3}x > \frac{1}{3}
To isolate the variable term on the left, we subtract 12\frac{1}{2} (which is equivalent to 36\frac{3}{6}) from 56\frac{5}{6} to get 26=13\frac{2}{6} = \frac{1}{3}.
3
Multiply both sides of the inequality by −32-\frac{3}{2} and reverse the inequality sign.
x<−12x < -\frac{1}{2}
Multiplying both sides of an inequality by a negative number requires reversing the direction of the inequality sign from >> to << to keep the inequality true.

Key Concept

Solving multi-step linear inequalities involving multiplication or division by a negative number.
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Topics

Solving Linear Inequalities

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