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GED Math: Linear Inequalities Concepts

Importance ★★★☆☆ · 10% (estimate)

Overview

This chapter covers linear inequalities in one variable: solving them, showing their solutions on a number line, and writing them from real situations with words such as "at least" and "at most."

An inequality is solved with the same steps as an equation, with one extra rule: when both sides are multiplied or divided by a negative number, the inequality sign flips. On a number line, a closed circle means the endpoint is included (≤ or ≥) and an open circle means it is not (< or >). Shading to the right shows greater values and shading to the left shows smaller values.

In word problems, "at most" and "no more than" mean ≤, while "at least" and "no less than" mean ≥. When the answer must be a whole number, such as a number of guests, round in the direction that keeps the inequality true.

Key concepts

  • Flipping the sign

    Multiplying or dividing both sides by a negative number reverses the inequality: < becomes >, and ≤ becomes ≥.

    Memory tipAdding or subtracting a negative number does not flip the sign.

  • Number line graphs

    Closed circle: the endpoint is included (≤, ≥). Open circle: the endpoint is not included (<, >). Shade the side that contains the solutions.

    Memory tipx ≤ 3 has a closed circle at 3 and shading to the left.

  • Key words

    "At most," "no more than," "maximum": ≤. "At least," "no less than," "minimum": ≥. "More than": >. "Less than": <.

    Memory tipPicture the limit: an elevator limit is a ceiling, so use ≤.

  • Testing a value

    Substitute a number into the original inequality. If the statement is true, the number is a solution.

    Memory tipTest a value on each side of your boundary to check the direction.

Often tested

  • Solve like an equation, then decide whether the sign flips.
  • For "greatest number of guests," round down so the budget is not exceeded.
  • For "least score needed," use the total needed minus the points already earned.
  • Check the endpoint: it works only with ≤ or ≥.

Easy to confuse: Open and closed circles

SymbolCircleEndpoint included?
< or >OpenNo
≤ or ≥Closed (filled in)Yes

3 sample questions

Verified practice questions from this unit.

Practice questionReviewedAlgebraic Problem Solving › Linear Inequalities★★★★☆

Question 1. Solve: −3x ≤ 12

  1. ① x ≤ −4
  2. ② x ≤ 4
  3. ③ x ≥ 4
  4. ④ x ≥ −4
▼ Show answer and explanation▲ Hide answer and explanation

Answer: ④ x ≥ −4

Key point: Dividing by a negative number flips the inequality sign

  1. Divide both sides by −3.
  2. Because you divide by a negative number, flip the sign: ≤ becomes ≥.
  3. 12 ÷ (−3) = −4, so x ≥ −4.

Check: Try x = 0: −3(0) = 0, and 0 ≤ 12 is true. 0 is greater than −4, so it fits x ≥ −4.

Wrong choices

  • x ≤ −4: x ≤ −4 forgets to flip the sign; x = −5 gives 15, which is not ≤ 12.
  • x ≥ 4: x ≥ 4 drops the negative sign from the answer; x = 0 works but is not ≥ 4.
  • x ≤ 4: x ≤ 4 has the wrong number and the wrong direction.
Practice questionReviewedAlgebraic Problem Solving › Linear Inequalities★★★☆☆

Question 2. Solve: 2x + 5 > 17

  1. ① x > 11
  2. ② x > 12
  3. ③ x < 6
  4. ④ x > 6
▼ Show answer and explanation▲ Hide answer and explanation

Answer: ④ x > 6

Key point: Solve an inequality like an equation (same steps)

  1. Subtract 5 from both sides: 2x > 12.
  2. Divide both sides by 2: x > 6.

Check: Try x = 7: 2(7) + 5 = 19, which is greater than 17. Try x = 6: 17 is not greater than 17, so 6 itself is not included.

Wrong choices

  • x < 6: x < 6 flips the sign, but you only flip when multiplying or dividing by a negative number.
  • x > 11: x > 11 adds 5 instead of subtracting it: (17 + 5) ÷ 2.
  • x > 12: x > 12 forgets to divide by 2.
Practice questionReviewedAlgebraic Problem Solving › Linear Inequalities★★★☆☆

Question 3. A freight elevator can carry at most 2,000 pounds. A worker who weighs 180 pounds rides with b boxes that weigh 40 pounds each. Which inequality shows the safe number of boxes?

  1. ① 180 + 40b < 2,000
  2. ② 40 + 180b ≤ 2,000
  3. ③ 180 + 40b ≤ 2,000
  4. ④ 180 + 40b ≥ 2,000
▼ Show answer and explanation▲ Hide answer and explanation

Answer: ③ 180 + 40b ≤ 2,000

Key point: "At most" means less than or equal to (≤)

  1. Total weight = worker + boxes = 180 + 40b.
  2. "At most 2,000" means the total can be 2,000 or less: ≤ 2,000.
  3. Inequality: 180 + 40b ≤ 2,000.

Check: With 45 boxes the total is exactly 180 + 1,800 = 1,980 pounds, which is allowed.

Wrong choices

  • 180 + 40b ≥ 2,000: ≥ means "at least," which would require the load to be 2,000 pounds or more.
  • 180 + 40b < 2,000: < leaves out exactly 2,000 pounds, but "at most" allows 2,000.
  • 40 + 180b ≤ 2,000: 40 + 180b switches the numbers, as if each box weighed 180 pounds.
Practice 5 questions from this unit →

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