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Difficulty: MediumSolving One-Variable Inequalities

What is the minimum integer value of xx that satisfies the inequality shown below?

3(2x−5)−8x≤4(x−1)−23 3(2x - 5) - 8x \leq 4(x - 1) - 23

Answer: 2

Answer

2
Solving the inequality yields x≥2x \geq 2. Since xx must be greater than or equal to 2, the minimum integer value that makes this statement true is exactly 2.

Step-by-Step Solution

1
Distribute the constants into the parentheses on both sides of the inequality.
6x−15−8x≤4x−4−236x - 15 - 8x \leq 4x - 4 - 23
This removes the parentheses, allowing us to combine like terms.
2
Combine the like terms on each side of the inequality.
−2x−15≤4x−27-2x - 15 \leq 4x - 27
Simplifying each side makes it easier to isolate the variable.
3
Subtract 4x4x from both sides to group the variable terms on the left side.
−6x−15≤−27-6x - 15 \leq -27
We need all terms with xx on one side of the inequality to solve for it.
4
Add 1515 to both sides to isolate the term with the variable.
−6x≤−12-6x \leq -12
This leaves only the −6x-6x term on the left side.
5
Divide both sides by −6-6, remembering to flip the inequality symbol.
x≥2x \geq 2
Dividing or multiplying an inequality by a negative number reverses the direction of the inequality sign.
6
Determine the minimum integer value that satisfies the simplified inequality.
2
The inequality states that xx must be greater than or equal to 22. The smallest integer that meets this condition is 22.

Key Concept

Solving linear inequalities requiring the distributive property, combining like terms, and reversing the inequality symbol when dividing by a negative coefficient.
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Topics

Solving One-Variable Inequalities

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