Inequalities • Topic 1 of 4

Solving Linear Inequalities

Solving a linear inequality uses the same steps as a linear equation: simplify each side, collect the variable on one side and constants on the other, then divide by the coefficient. As long as you only add, subtract, or multiply/divide by a positive number, the inequality sign stays exactly as it is. The answer is a range, not a single number — 2x + 3 < 11 gives x < 4, meaning every value below 4 works. It helps to picture the solution on a number line: an open circle for strict < or >, a filled circle for ≤ or ≥, with the shading showing the direction. On the GRE you rarely graph, but knowing whether the boundary is included (≤) or excluded (<) matters when a question asks for the largest or smallest integer that satisfies the condition.

A number line for x greater than 2, with an open circle at 2 and the line shaded to the right toward larger valuesx > 2-5-4-3-2-1012345Open circle: x = 2 is not included.

✅ Solved examples

1. Solve 2x + 3 < 11.
Subtract 3: 2x < 8. Divide by 2 (positive, sign stays): x < 4.
2. Solve 5x − 7 ≥ 2x + 5.
Collect: 3x ≥ 12. Divide by 3: x ≥ 4. (Boundary 4 is included.)
3. Solve 3(x − 2) ≤ x + 4.
Expand: 3x − 6 ≤ x + 4. Collect: 2x ≤ 10, so x ≤ 5.
4. What is the largest integer x with 4x + 1 < 21?
4x < 20 ⇒ x < 5. Since x < 5 (not ≤), the largest integer is 4.

✏️ Practice — try these, take hints as needed

1. Solve 3x − 4 > 8.
Add 4.
3x > 12.
Divide by 3, positive.
x > 4
2. Solve 2x + 9 ≤ 5x − 3.
Collect variables.
12 ≤ 3x.
Divide by 3.
x ≥ 4
3. Solve 4(x + 1) < 2x + 10.
Distribute.
4x + 4 < 2x + 10.
2x < 6.
x < 3
4. What is the smallest integer x with 3x − 2 > 10?
3x > 12.
x > 4.
Strict, so not 4.
5
5. Solve 7 − x ≥ 2 without dividing by a negative.
Add x to both sides.
7 ≥ 2 + x.
Subtract 2.
x ≤ 5

📝 Topic test — 8 questions

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