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Solving Linear Inequalities

Solve one-variable inequalities step by step, including when to flip the sign.

Practice this skill 22 min Solve linear inequalities Leer en español
Key idea

What you'll learn

An inequality is like an equation, but instead of saying two sides are equal, it says one side is bigger or smaller than the other. In this lesson you will learn to solve one-variable linear inequalities such as 3x - 5 < 16 or 5 - 2x > 11.

Here is the good news: you already know almost everything you need. You solve an inequality the same way you solve an equation — undo addition with subtraction, undo multiplication with division, and keep the two sides balanced. There is just one new rule: when you multiply or divide both sides by a negative number, you must flip the inequality sign.

Why it matters on the GED: inequalities appear in several forms on the Mathematical Reasoning test:

  • "Which of the following is the solution to -4x \le 20?" (multiple choice)
  • "Select all values of x that make the inequality true." (select-all)
  • "Type the boundary value." (fill-in-the-blank)
  • Word problems about budgets, weight limits and scores (covered in the next lessons).

In real life, inequalities answer questions like "How many hours can I work and still keep my benefits?" or "How many items can I buy with the money I have?" — whenever the answer is a range of numbers instead of one number.

Words to know

Key words

  • Inequality — a math sentence that compares two amounts using <, >, \le, or \ge (example: x + 2 > 7).
  • < (less than) — the left side is smaller (example: 3 < 8).
  • > (greater than) — the left side is bigger (example: 8 > 3).
  • \le (less than or equal to) — smaller or the same (example: 5 \le 5 is true).
  • \ge (greater than or equal to) — bigger or the same (example: 6 \ge 2 is true).
  • Variable — a letter that stands for an unknown number (example: x).
  • Coefficient — the number multiplying the variable (in -4x, the coefficient is -4).
  • Solution — a value of the variable that makes the inequality true (example: x = 6 is a solution of x + 2 > 7 because 8 > 7).
  • Solution set — all the solutions together, usually written like x > 5.
  • Boundary value — the number where the solution set starts or stops (in x > 5, the boundary is 5).
  • Inverse operation — the operation that undoes another (subtraction undoes addition; division undoes multiplication).
  • Reverse (flip) the sign — change < to >, > to <, \le to \ge, or \ge to \le.
  • Distribute — multiply a number into parentheses: 2(x + 4) = 2x + 8.

All Math words in the glossary

Step 1: Solve it like an equation

An inequality stays true when you do the same thing to both sides, as long as that "thing" is:

  • adding any number,
  • subtracting any number,
  • multiplying or dividing by a positive number.

So most inequalities are solved exactly like equations.

Example A. Solve x + 7 > 12.

Subtract 7 from both sides: x + 7 - 7 > 12 - 7, so x > 5.

Number line from 0 to 10. Open circle at 5 with shading and an arrow to the right: the graph of x > 5. x + 7 > 12 → x > 5 0 1 2 3 4 5 6 7 8 9 10

Example B. Solve 3x - 5 < 16.

  1. Add 5 to both sides: 3x < 21.
  2. Divide both sides by 3 (a positive number — no flip): x < 7.
Number line from 2 to 12. Open circle at 7 with shading and an arrow to the left: the graph of x < 7. 3x − 5 < 16 → x < 7 2 3 4 5 6 7 8 9 10 11 12

Example C (decimals). Solve 0.5x + 1.25 \ge 3.75.

  1. Subtract 1.25: 0.5x \ge 2.5.
  2. Divide by 0.5: x \ge 5.
Number line from 0 to 10. Closed circle at 5 with shading and an arrow to the right: the graph of x >= 5. 0.5x + 1.25 ≥ 3.75 → x ≥ 5 0 1 2 3 4 5 6 7 8 9 10

Example D (fractions). Solve \tfrac{2}{3}x - 1 < 7.

  1. Add 1: \tfrac{2}{3}x < 8.
  2. Multiply by the reciprocal \tfrac{3}{2} (positive — no flip): x < 8 \cdot \tfrac{3}{2} = 12.
Number line from 4 to 16. Open circle at 12 with shading and an arrow to the left: the graph of x < 12. (2/3)x − 1 < 7 → x < 12 4 5 6 7 8 9 10 11 12 13 14 15 16

Notice that a negative constant does not cause a flip. In 2x - 9 > -3, you add 9 to get 2x > 6, then divide by positive 2 to get x > 3. Only multiplying or dividing by a negative flips the sign.

Step 2: The flip rule (multiplying or dividing by a negative)

Start with a true statement: 2 < 5.

Now multiply both sides by -1: you get -2 and -5. On the number line, -2 is to the right of -5, so -2 is bigger: -2 > -5. The order switched! That is why the rule exists:

Number line from -6 to 6. Blue dots at 2 and 5 and red dots at -2 and -5, with dashed arcs showing that multiplying by -1 mirrors each number across 0. Before: 2 < 5. After: -2 > -5, because -2 is to the right of -5. The order switched, so the inequality sign must flip. Multiply both sides of 2 < 5 by −1 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 ×(−1) mirrors each number across 0 before: 2 < 5 after: −2 > −5 The order switched, so the sign must flip.

When you multiply or divide both sides of an inequality by a negative number, reverse the direction of the inequality sign.

Original signAfter multiplying/dividing by a negative
<>
><
\le\ge
\ge\le

Example E. Solve -4x \le 20.

Divide both sides by -4 and flip: x \ge \dfrac{20}{-4}, so x \ge -5.

Number line from -10 to 0. Closed circle at -5 with shading and an arrow to the right: the graph of x >= -5. −4x ≤ 20 → divide by −4, flip → x ≥ −5 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0

Example F. Solve 5 - 2x > 11.

  1. Subtract 5: -2x > 6.
  2. Divide by -2 and flip: x < -3.

Example G. Solve \dfrac{x}{-3} + 2 > 5.

  1. Subtract 2: \dfrac{x}{-3} > 3.
  2. Multiply by -3 and flip: x < -9.
Number line from -14 to -4. Open circle at -9 with shading and an arrow to the left: the graph of x < -9. x/(−3) + 2 > 5 → x < −9 −14 −13 −12 −11 −10 −9 −8 −7 −6 −5 −4

Avoid the flip completely (optional trick). You can keep the variable term positive by moving it to the other side. For 5 - 2x > 11: add 2x to both sides to get 5 > 11 + 2x, subtract 11 to get -6 > 2x, divide by positive 2 to get -3 > x. Reading it from the right, -3 > x means the same as x < -3. Same answer, no flip needed.

Number line from -6 to 2. Open circle at -3 with shading and an arrow to the left: the graph of x < -3. −6 −5 −4 −3 −2 −1 0 1 2

Step 3: Parentheses and variables on both sides

GED-hard inequalities combine several steps. Use this order every time:

  1. Distribute to remove parentheses. Multiply the outside number by every term inside, signs included: -3(x - 2) = -3x + 6.
  2. Combine like terms on each side.
  3. Collect the variable terms on one side (add or subtract them).
  4. Collect the constants on the other side.
  5. Divide by the coefficient. Flip the sign only if that coefficient is negative.
  6. Check with a test value.

Example H. Solve 2(x + 4) \ge 3x - 1.

  1. Distribute: 2x + 8 \ge 3x - 1.
  2. Subtract 2x: 8 \ge x - 1.
  3. Add 1: 9 \ge x, which means x \le 9.
Number line from 4 to 14. Closed circle at 9 with shading and an arrow to the left: the graph of x <= 9. 9 ≥ x is the same as x ≤ 9 4 5 6 7 8 9 10 11 12 13 14

Example I. Solve 4 - 3(x - 2) > 2x - 15.

  1. Distribute -3: 4 - 3x + 6 > 2x - 15.
  2. Combine: 10 - 3x > 2x - 15.
  3. Subtract 2x: 10 - 5x > -15.
  4. Subtract 10: -5x > -25.
  5. Divide by -5 and flip: x < 5.

How to check any answer. Pick an easy number inside your solution set and one outside it. In Example I, try x = 0 (inside): 4 - 3(-2) = 10 and 2(0) - 15 = -15; is 10 > -15? Yes. Try x = 10 (outside): 4 - 3(8) = -20 and 2(10) - 15 = 5; is -20 > 5? No. Both tests agree, so x < 5 is right.

Number line from -2 to 12. Open circle at 5 with shading and an arrow to the left: the graph of x < 5. Test value 0 is marked true (in the solution set). Test value 10 is marked false (not a solution). 4 − 3(x − 2) > 2x − 15 → x < 5 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 x = 0 ✓ x = 10 ✗

Greatest/least whole number questions. Sometimes the GED asks for "the greatest integer that satisfies…". Solve first, then look at the boundary. If x \le \tfrac{25}{6} \approx 4.17, the greatest integer is 4. If x < 4 (strict), the greatest integer is 3, because 4 itself is not allowed.

Two number lines from 2 to 6. Top: closed circle at 25/6, about 4.17, shaded left; the integer 4 is highlighted as the greatest integer solution. Bottom: open circle at 4, shaded left, for the strict inequality x < 4; 4 is not allowed and 3 is highlighted as the greatest integer solution. x ≤ 25/6 ≈ 4.17 → greatest integer 4 2 3 4 5 6 4.17 4 is the greatest integer x < 4 (strict) → greatest integer 3 2 3 4 5 6 4 not allowed 3 is the greatest integer
Formula

Rules to memorize (not on the formula sheet)

There is no formula to plug into, but these properties of inequality must be memorized — they are not on the GED formula sheet.

For any numbers a, b, c:

\text{If } a < b, \text{ then } a + c < b + c \quad\text{and}\quad a - c < b - c \text{If } a < b \text{ and } c > 0, \text{ then } ac < bc \quad\text{and}\quad \frac{a}{c} < \frac{b}{c} \text{If } a < b \text{ and } c < 0, \text{ then } ac > bc \quad\text{and}\quad \frac{a}{c} > \frac{b}{c}
  • a, b are the two sides of the inequality.
  • c is the number you add, subtract, multiply or divide by.
  • The same rules hold for >, \le and \ge.

Also remember: a < b means exactly the same thing as b > a. You may always read an inequality from right to left as long as you turn the symbol around with it (the "mouth" of the symbol still opens toward the larger side).

Worked example

Worked example 1 (easy): one step

Problem: Solve x - 8 \le 3.

  1. Undo the subtraction by adding 8 to both sides: x - 8 + 8 \le 3 + 8.
  2. Simplify: x \le 11.

Answer: x \le 11

Number line from 6 to 16. Closed circle at 11 with shading and an arrow to the left: the graph of x <= 11. Test value 11 is marked true (in the solution set). Test value 12 is marked false (not a solution). x − 8 ≤ 3 → x ≤ 11 6 7 8 9 10 11 12 13 14 15 16 x = 11 ✓ x = 12 ✗

Check: Try x = 11: 11 - 8 = 3, and 3 \le 3 is true. Try x = 12 (outside): 12 - 8 = 4, and 4 \le 3 is false. ✓

Worked example

Worked example 2: dividing by a negative

Problem: Solve 7 - 3x \ge 22.

  1. Subtract 7 from both sides: -3x \ge 15.
  2. Divide both sides by -3. The divisor is negative, so flip \ge to \le: x \le \dfrac{15}{-3}.
  3. Simplify: x \le -5.

Answer: x \le -5

Number line from -10 to 2. Closed circle at -5 with shading and an arrow to the left: the graph of x <= -5. Test value -6 is marked true (in the solution set). Test value 0 is marked false (not a solution). 7 − 3x ≥ 22 → divide by −3, flip → x ≤ −5 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 2 x = −6 ✓ x = 0 ✗

Check: Try x = -6: 7 - 3(-6) = 7 + 18 = 25, and 25 \ge 22 is true. Try x = 0 (outside): 7 - 0 = 7, and 7 \ge 22 is false. ✓

Worked example

Worked example 3 (real world): a phone budget

Problem: A phone plan costs $35 per month plus $4 for each gigabyte (GB) of extra data. Dana wants her monthly bill to be no more than $63. Solve 35 + 4g \le 63 to find how many extra gigabytes she can use.

  1. Subtract 35 from both sides: 4g \le 28.
  2. Divide by 4 (positive, no flip): g \le 7.

Answer: Dana can use at most 7 extra GB (g \le 7).

Bar model of Dana's phone bill: a 35 dollars plan block followed by seven 4 dollars blocks for 7 extra gigabytes, totaling 28 dollars and reaching the 63 dollars limit line exactly. A dashed red eighth 4 dollars block goes past the limit to 67 dollars, which is too much. So 35 + 4g <= 63 gives g <= 7. 35 + 4g ≤ 63 $35 plan limit $63 7 GB × $4 = $28 8th GB: $67 ✗

Check: 35 + 4(7) = 35 + 28 = 63, which is exactly the limit. With 8 GB: 35 + 32 = 67 > 63, too much. ✓

Worked example

Worked example 4 (GED-hard): parentheses on both sides

Problem: Solve 3(2 - x) + 4 < 5(x - 2).

  1. Distribute on the left: 6 - 3x + 4 < 5(x - 2).
  2. Distribute on the right: 6 - 3x + 4 < 5x - 10.
  3. Combine the left side: 10 - 3x < 5x - 10.
  4. Subtract 5x from both sides: 10 - 8x < -10.
  5. Subtract 10 from both sides: -8x < -20.
  6. Divide by -8 and flip: x > \dfrac{-20}{-8} = 2.5.

Answer: x > 2.5

Number line from 0 to 5. Open circle at 2.5 with shading and an arrow to the right: the graph of x > 2.5. Test value 3 is marked true (in the solution set). Test value 2 is marked false (not a solution). 3(2 − x) + 4 < 5(x − 2) → x > 2.5 0 1 2 3 4 5 2.5 x = 3 ✓ x = 2 ✗

Check: Try x = 3: left side 3(-1) + 4 = 1; right side 5(1) = 5; 1 < 5 ✓. Try x = 2 (outside): left 3(0) + 4 = 4; right 5(0) = 0; 4 < 0 is false ✓.

Worked example

Worked example 5 (work backwards): the greatest integer

Problem: What is the greatest integer x that makes \dfrac{x}{4} - 3 < 2 true?

  1. Add 3 to both sides: \dfrac{x}{4} < 5.
  2. Multiply both sides by 4 (positive, no flip): x < 20.
  3. The sign is strict (<), so 20 itself is not a solution. The greatest integer below 20 is 19.

Answer: 19

Number line from 14 to 22. Open circle at 20 with shading and an arrow to the left: the graph of x < 20. The integer 19 is marked as the greatest integer solution. x/4 − 3 < 2 → x < 20 14 15 16 17 18 19 20 21 22 19 = greatest integer 20 ✗

Check: \dfrac{19}{4} - 3 = 4.75 - 3 = 1.75 < 2 ✓. And \dfrac{20}{4} - 3 = 2, but 2 < 2 is false, so 20 fails. ✓

Calculator tip

Calculator tip (TI-30XS MultiView)

Most inequality items are short enough to do by hand, and simple ones (like x + 7 > 12) often appear in the calculator-free part of the test. Use the calculator only for messy arithmetic such as decimals or fractions.

  • Type a negative number with the (−) key, not the minus key. For \dfrac{15}{-3} press 15 ÷ (−) 3 enter → -5.
  • For fractions use the n/d key. Press toggle to switch the answer between a fraction and a decimal (for example, \tfrac{25}{6} \leftrightarrow 4.1666\ldots).
  • To check a solution, type the left side with your test value and compare it with the right side yourself.

What the calculator cannot do: it does not know about the flip rule. It will happily compute 20 \div (-4) = -5, but you must remember to turn \le into \ge. Write the sign on your erasable note board before you press enter.

Common mistake

Common mistakes

  • Forgetting to flip. -4x \le 20 → ✗ x \le -5. ✓ Dividing by -4 flips the sign: x \ge -5.
  • Flipping when you shouldn't. 2x > -6 → ✗ x < -3. ✓ You divided by positive 2, so no flip: x > -3. A negative number on the right side never causes a flip; only a negative multiplier or divisor does.
  • Using the wrong inverse operation. x + 7 > 12 → ✗ x > 19 (added 7). ✓ Subtract 7: x > 5.
  • Distributing to only the first term. -3(x - 2) → ✗ -3x - 2 or -3x - 6. ✓ -3x + 6 (a negative times a negative is positive).
  • Changing < into \le (or the reverse). The "or equal to" part never appears or disappears while you solve. If you start with <, the answer uses < or >.
  • Reading 9 \ge x backwards. 9 \ge x means "x is at most 9", i.e. x \le 9 — not x \ge 9.
Test strategy

GED test strategy

  • Read the question twice. Does it ask for the solution set, a single value that works, or the greatest/least integer?
  • Look at the signs in the answer choices first. Often two choices have the same number but opposite signs (x \ge -5 vs x \le -5). That tells you the question is testing the flip rule — slow down at the division step.
  • Plug in a test value. Choose an easy number from one answer choice (0 is great when allowed) and substitute it into the original inequality. If it makes the inequality false, cross off every choice that includes that number.
  • Check the boundary. Substitute the boundary value: the two sides should be equal. If they are not, the number is wrong.
  • Eliminate by direction. If x = 100 obviously works in the original, the answer must point to the right (> or \ge).
  • Budget your time. A one-step inequality should take under a minute. Spend extra time only on multi-step items with parentheses.

Check yourself

Solve linear inequalitiesNo calculator

Solve the inequality x + 7 > 12.

Answer choices
Hint

Undo the +7 with the opposite operation. Does adding or subtracting a number ever change the direction of the sign?

Check yourself

Solve linear inequalitiesNo calculator

Solve the inequality -4x \le 20.

Answer choices
Hint

To get x alone you must divide by a negative number. What happens to the inequality sign when you do that?

Summary

  • Solve an inequality like an equation: distribute, combine like terms, collect variables on one side and constants on the other.
  • Adding, subtracting, or multiplying/dividing by a positive number keeps the sign the same.
  • Multiplying or dividing by a negative number flips the sign: < ↔ >, \le ↔ \ge.
  • 9 \ge x is the same as x \le 9 — turn the whole statement around together.
  • Always check with one value inside and one outside your solution set.
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