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Difficulty: MediumSolving Linear Inequalities

For all real values of xx that satisfy the inequality 5−3x−24≥x+625 - \frac{3x - 2}{4} \ge \frac{x + 6}{2}, the solution set is represented by x≤bx \le b. What is the value of bb?

Answer: 2

Answer

The value of bb is 2.
To solve 5−3x−24≥x+625 - \frac{3x - 2}{4} \ge \frac{x + 6}{2}, multiply all terms by 4 to clear the denominators, resulting in 20−(3x−2)≥2(x+6)20 - (3x - 2) \ge 2(x + 6). Carefully distribute the negative sign to obtain 20−3x+2≥2x+1220 - 3x + 2 \ge 2x + 12. Simplifying the left side gives 22−3x≥2x+1222 - 3x \ge 2x + 12. Moving the variable terms to one side yields 10≥5x10 \ge 5x, which simplifies to x≤2x \le 2. Thus, the upper bound value bb is 2.

Step-by-Step Solution

1
Multiply both sides of the inequality by the least common denominator, which is 4.
20−(3x−2)≥2(x+6)20 - (3x - 2) \ge 2(x + 6)
Multiplying all terms by the common denominator eliminates fractions and simplifies the inequality.
2
Distribute the negative sign to the numerator terms on the left and distribute the 2 on the right.
20−3x+2≥2x+1220 - 3x + 2 \ge 2x + 12
Distributing the negative sign across (3x−2)(3x - 2) changes it to −3x+2-3x + 2. Distributing 2 across (x+6)(x + 6) yields 2x+122x + 12.
3
Combine like terms on the left side of the inequality.
22−3x≥2x+1222 - 3x \ge 2x + 12
Combining the constant terms 2020 and 22 simplifies the expression to 2222.
4
Isolate the variable terms by adding 3x3x and subtracting 12 from both sides.
10≥5x10 \ge 5x
Grouping variables on one side and constants on the other allows us to solve for xx.
5
Divide both sides by 5.
2≥x2 \ge x (or x≤2x \le 2)
Dividing by a positive number isolates the variable without changing the direction of the inequality sign.

Key Concept

Solving linear inequalities involving fractions and distributing negative coefficients.

Alternative Method

We can write the inequality by separating each fraction term first: 5−34x+24≥12x+625 - \frac{3}{4}x + \frac{2}{4} \ge \frac{1}{2}x + \frac{6}{2}. This simplifies to 5.5−0.75x≥0.5x+35.5 - 0.75x \ge 0.5x + 3. Subtracting 0.5x0.5x and 5.55.5 from both sides gives −1.25x≥−2.5-1.25x \ge -2.5. Dividing by −1.25-1.25 and reversing the inequality sign gives x≤2x \le 2.
Estimated Time:1m 30s
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Topics

Solving Linear Inequalities

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