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Difficulty: Very hardSolving One-Variable Inequalities

What is the complete solution set for the given inequality?

12(−8x−10)−23(9x+15)≤−4(x−2)+2x+1 \frac{1}{2}(-8x - 10) - \frac{2}{3}(9x + 15) \leq -4(x - 2) + 2x + 1
  1. x≥−3x \geq -3Answer
  2. B
    x≤−3x \leq -3
  3. C
    x≥−1x \geq -1
  4. D
    x≥−2x \geq -2

Answer

The correct solution is x≥−3x \geq -3.
By systematically distributing the fractions and negative integers on both sides, we simplify the expression to −10x−15≤−2x+9-10x - 15 \leq -2x + 9. Moving the variable terms to the left and constants to the right results in −8x≤24-8x \leq 24. Finally, dividing by −8-8 isolates xx and requires flipping the inequality symbol, providing the correct solution.

Step-by-Step Solution

1
Distribute the fractions into the terms inside the parentheses on the left side.
(−4x−5)−(6x+10)≤−4(x−2)+2x+1(-4x - 5) - (6x + 10) \leq -4(x - 2) + 2x + 1
To eliminate the parentheses and clear the fractions for easier manipulation.
2
Distribute the −4-4 on the right side of the inequality.
−4x−5−6x−10≤−4x+8+2x+1-4x - 5 - 6x - 10 \leq -4x + 8 + 2x + 1
To expand all terms fully before attempting to combine them.
3
Combine the like terms on both the left and right sides independently.
−10x−15≤−2x+9-10x - 15 \leq -2x + 9
To simplify the inequality into a standard two-step format.
4
Add 2x2x to both sides and add 1515 to both sides to group variable terms and constant terms.
−8x≤24-8x \leq 24
To isolate the xx term on one side of the inequality.
5
Divide both sides by −8-8 and reverse the direction of the inequality symbol.
x≥−3x \geq -3
Multiplying or dividing an inequality by a negative number requires flipping the inequality sign to maintain the mathematical truth.

Key Concept

Solving Complex Linear Inequalities
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Topics

Solving One-Variable Inequalities

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