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Difficulty: MediumLinear Inequalities in One Variable

What is the solution set for the inequality 5−2(3x−1)≥−4x+115 - 2(3x - 1) \geq -4x + 11?

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

Answer

The correct solution is xx is less than or equal to −2-2.
To solve the inequality, first distribute the negative two to the terms inside the parentheses to get 5−6x+2≥−4x+115 - 6x + 2 \geq -4x + 11. Combining the constants on the left side yields 7−6x≥−4x+117 - 6x \geq -4x + 11. Adding four xx to both sides results in 7−2x≥117 - 2x \geq 11, and subtracting seven from both sides isolates the variable term, giving −2x≥4-2x \geq 4. Dividing both sides by negative two and reversing the inequality sign yields the correct solution, xx is less than or equal to negative two.

Step-by-Step Solution

1
Distribute the −2-2 to the terms inside the parentheses.
5−6x+2≥−4x+115 - 6x + 2 \geq -4x + 11
To simplify the expression by removing the parentheses.
2
Combine the constant terms on the left side of the inequality.
7−6x≥−4x+117 - 6x \geq -4x + 11
To group like terms together before isolating the variable.
3
Add 4x4x to both sides of the inequality.
7−2x≥117 - 2x \geq 11
To move all variable terms to one side of the inequality.
4
Subtract 77 from both sides of the inequality.
−2x≥4-2x \geq 4
To isolate the variable term.
5
Divide both sides by −2-2 and reverse the inequality sign.
x≤−2x \leq -2
Dividing an inequality by a negative number requires reversing the direction of the inequality sign to keep the statement true.

Key Concept

Solving linear inequalities in one variable using the distributive property and applying the sign reversal rule when dividing by a negative number.
Estimated Time:1m 30s
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Topics

Linear Inequalities in One Variable

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