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Difficulty: HardSolving One-Variable Inequalities

For the inequality below, which of the following expresses all possible values of yy?

2(3y−5)−3(y−4)≥6y−162(3y - 5) - 3(y - 4) \geq 6y - 16
  1. y≤6y \leq 6Answer
  2. B
    y≥6y \geq 6
  3. C
    y≤−2y \leq -2
  4. D
    y≥−2y \geq -2

Answer

The correct answer is y≤6y \leq 6.
The correct answer is derived by distributing the terms to get 6y−10−3y+12≥6y−166y - 10 - 3y + 12 \geq 6y - 16, which simplifies to 3y+2≥6y−163y + 2 \geq 6y - 16. Subtracting 6y6y from both sides and subtracting 2 from both sides yields −3y≥−18-3y \geq -18. Finally, dividing by -3 requires flipping the inequality symbol, resulting in y≤6y \leq 6.

Step-by-Step Solution

1
Distribute the constants on the left side of the inequality.
6y−10−3y+12≥6y−166y - 10 - 3y + 12 \geq 6y - 16
To clear the parentheses. Note that multiplying -3 by -4 results in a positive 12.
2
Combine the like terms on the left side.
3y+2≥6y−163y + 2 \geq 6y - 16
To simplify the expression before isolating the variable.
3
Move all variable terms to the left side and constant terms to the right side.
3y−6y≥−16−23y - 6y \geq -16 - 2, which simplifies to −3y≥−18-3y \geq -18
To group the unknown variable on one side of the inequality.
4
Divide both sides by -3.
y≤6y \leq 6
To solve for y. Whenever multiplying or dividing an inequality by a negative number, the direction of the inequality sign must be reversed.

Key Concept

Solving One-Variable Inequalities
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Topics

Solving One-Variable Inequalities

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