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Difficulty: MediumLinear Inequalities in One Variable

What is the solution set for the inequality −2(3x−4)+5≤−x+3-2(3x - 4) + 5 \leq -x + 3?

  1. A
    x≤2x \leq 2
  2. B
    x≥−65x \geq -\frac{6}{5}
  3. x≥2x \geq 2Answer
  4. D
    x≥107x \geq \frac{10}{7}

Answer

x≥2x \geq 2
Distributing −2-2 over 3x−43x - 4 yields −6x+8-6x + 8. Combining the constants on the left gives −6x+13≤−x+3-6x + 13 \leq -x + 3. Subtracting 1313 and adding xx to both sides results in −5x≤−10-5x \leq -10. Dividing both sides by −5-5 and flipping the inequality symbol yields the solution set of all values greater than or equal to 22.

Step-by-Step Solution

1
Distribute the −2-2 coefficient on the left side of the inequality.
−6x+8+5≤−x+3-6x + 8 + 5 \leq -x + 3
Applying the distributive property simplifies the expression inside the parentheses.
2
Combine the constant terms on the left side of the inequality.
−6x+13≤−x+3-6x + 13 \leq -x + 3
Combining like terms simplifies the left-hand side before isolating the variable.
3
Subtract 1313 from both sides of the inequality.
−6x≤−x−10-6x \leq -x - 10
This begins the process of isolating the variable terms on one side and constant terms on the other.
4
Add xx to both sides of the inequality.
−5x≤−10-5x \leq -10
This groups all variable terms together on the left-hand side.
5
Divide both sides of the inequality by −5-5 and reverse the direction of the inequality sign.
x≥2x \geq 2
Dividing an inequality by a negative number requires reversing the inequality sign to maintain a true statement.

Key Concept

Solving linear inequalities in one variable using algebraic properties, specifically distributing a negative value and reversing the inequality direction when dividing by a negative number.
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Topics

Linear Inequalities in One Variable

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