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Difficulty: MediumLinear Inequalities in One Variable

Which of the following represents all possible values of xx that satisfy the inequality −3(2x−5)−4x≥35-3(2x - 5) - 4x \geq 35?

  1. A
    x≥−2x \geq -2
  2. x≤−2x \leq -2Answer
  3. C
    x≤−5x \leq -5
  4. D
    x≤2x \leq 2

Answer

The inequality is satisfied by all values of xx such that x≤−2x \leq -2.
To solve the inequality, distribute −3-3 across the terms in the parentheses to get −6x+15−4x≥35-6x + 15 - 4x \geq 35. Grouping the xx terms gives −10x+15≥35-10x + 15 \geq 35. Subtracting 15 from both sides yields −10x≥20-10x \geq 20. Finally, dividing both sides by −10-10 and reversing the inequality symbol gives x≤−2x \leq -2.

Step-by-Step Solution

1
Distribute the negative coefficient −3-3 to both terms inside the parentheses.
−6x+15−4x≥35-6x + 15 - 4x \geq 35
Applying the distributive property a(b+c)=ab+aca(b + c) = ab + ac allows the removal of parentheses to simplify the expression.
2
Combine the variable terms −6x-6x and −4x-4x on the left side of the inequality.
−10x+15≥35-10x + 15 \geq 35
Combining like terms simplifies the left side of the inequality to prepare for isolating the variable.
3
Isolate the variable term by subtracting 15 from both sides of the inequality.
−10x≥20-10x \geq 20
Using the subtraction property of inequality maintains the balance of the inequality while isolating the term with the variable.
4
Divide both sides of the inequality by −10-10 and reverse the direction of the inequality symbol.
x≤−2x \leq -2
Dividing both sides of an inequality by a negative number requires reversing the inequality sign to keep the inequality statement true.

Key Concept

Solving multi-step linear inequalities in one variable, including distributing terms and reversing the inequality direction when dividing by a negative number.
Estimated Time:1m 30s
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Topics

Linear Inequalities in One Variable

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