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Difficulty: HardSolving One-Variable Inequalities

Find the greatest integer value of xx that makes the following inequality true:

23(6x−9)−4(2x+1)>14(8x−20)−3x \frac{2}{3}(6x - 9) - 4(2x + 1) > \frac{1}{4}(8x - 20) - 3x

Answer: -2

Answer

The greatest integer value that satisfies the inequality is -2.
Expanding both sides yields −4x−10>−x−5-4x - 10 > -x - 5. Isolating xx on one side gives −3x>5-3x > 5. Dividing by -3 requires flipping the inequality symbol, resulting in x<−53x < -\frac{5}{3} (or x<−1.67x < -1.67). The greatest integer strictly less than -1.67 is -2.

Step-by-Step Solution

1
Distribute the constants and fractions into the parentheses on both sides.
4x−6−8x−4>2x−5−3x4x - 6 - 8x - 4 > 2x - 5 - 3x
Applying the distributive property removes the parentheses so like terms can be combined.
2
Combine like terms on each side of the inequality.
−4x−10>−x−5-4x - 10 > -x - 5
Simplifying both expressions makes it easier to isolate the variable.
3
Add xx to both sides and add 1010 to both sides to separate the variable terms from the constant terms.
−3x>5-3x > 5
This isolates the xx term on one side of the inequality.
4
Divide both sides by -3 and flip the inequality sign.
x<−53x < -\frac{5}{3}
When multiplying or dividing an inequality by a negative number, the direction of the inequality sign must be reversed.
5
Identify the greatest integer less than −53-\frac{5}{3}.
Since −53≈−1.67-\frac{5}{3} \approx -1.67, the integers strictly less than this are −2,−3,−4,…-2, -3, -4, \dots The greatest of these is −2-2.
The question specifically asks for the greatest integer value that makes the inequality true.

Key Concept

Solving complex linear inequalities involves applying the distributive property, combining like terms, and reversing the inequality symbol when dividing or multiplying by a negative number.
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Topics

Solving One-Variable Inequalities

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