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Difficulty: MediumLinear Inequalities in One Variable

What is the complete set of solutions for the inequality 7−3(2x−5)≤−4x+87 - 3(2x - 5) \leq -4x + 8?

  1. A
    x≤7x \leq 7
  2. B
    x≥−8x \geq -8
  3. x≥7x \geq 7Answer
  4. D
    x≥−7x \geq -7

Answer

The complete set of solutions is represented by the inequality x≥7x \geq 7.
To solve the inequality 7−3(2x−5)≤−4x+87 - 3(2x - 5) \leq -4x + 8, we first distribute −3-3 to get 7−6x+15≤−4x+87 - 6x + 15 \leq -4x + 8. Combining constant terms on the left side gives 22−6x≤−4x+822 - 6x \leq -4x + 8. Adding 4x4x and subtracting 2222 from both sides isolates the variable, resulting in −2x≤−14-2x \leq -14. Dividing both sides by −2-2 and flipping the inequality symbol yields the solution x≥7x \geq 7.

Step-by-Step Solution

1
Distribute the −3-3 to both terms inside the parentheses.
7−6x+15≤−4x+87 - 6x + 15 \leq -4x + 8
Applying the distributive property, −3(2x)=−6x-3(2x) = -6x and −3(−5)=15-3(-5) = 15.
2
Combine the constant terms on the left side of the inequality.
22−6x≤−4x+822 - 6x \leq -4x + 8
Adding 77 and 1515 yields 2222.
3
Isolate the variable term on one side and the constant term on the other side.
−2x≤−14-2x \leq -14
Adding 4x4x to both sides yields −2x-2x, and subtracting 2222 from both sides yields −14-14.
4
Divide both sides by −2-2 and flip the inequality sign.
x≥7x \geq 7
Dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol.

Key Concept

Solving multi-step linear inequalities in one variable including negative coefficient division and the distributive property.
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Topics

Linear Inequalities in One Variable

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