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Difficulty: HardSolving Linear Inequalities

What is the greatest integer value of xx that satisfies the inequality 2−5x3−x−42≥3\frac{2 - 5x}{3} - \frac{x - 4}{2} \geq 3?

Answer: -1

Answer

The greatest integer value of xx that satisfies the inequality is -1.
The correct answer is -1 because solving the inequality leads to x≤−213x \leq -\frac{2}{13}. Since −213-\frac{2}{13} is approximately −0.154-0.154, the set of integers satisfying the inequality is {−1,−2,−3,… }\{-1, -2, -3, \dots\}. The greatest integer in this set is -1.

Step-by-Step Solution

1
Multiply the entire inequality by the least common multiple of the denominators (6) to eliminate the fractions.
2(2−5x)−3(x−4)≥182(2 - 5x) - 3(x - 4) \geq 18
Multiplying by a positive number clears the fractions without changing the direction of the inequality.
2
Distribute the coefficients and combine like terms on the left side of the inequality.
16−13x≥1816 - 13x \geq 18
Simplifying the expressions on each side makes it easier to isolate the variable.
3
Subtract 16 from both sides to isolate the term with the variable xx.
−13x≥2-13x \geq 2
Moving the constant terms to one side prepares the inequality for division.
4
Divide both sides by -13 and reverse the direction of the inequality sign.
x≤−213x \leq -\frac{2}{13}
Dividing by a negative number requires flipping the inequality sign to maintain a true statement.
5
Determine the greatest integer that is less than or equal to −213-\frac{2}{13}.
-1
Since −213≈−0.154-\frac{2}{13} \approx -0.154, the largest integer that is less than or equal to this value is -1.

Key Concept

Solving multi-step linear inequalities, including clearing fractional coefficients and reversing the inequality sign when multiplying or dividing by a negative number.
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Topics

Solving Linear Inequalities

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