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Writing Inequalities from Word Problems

Translate everyday situations into one-variable linear inequalities.

Practice this skill 20 min Write inequalities from context Leer en español
Key idea

What you'll learn

On the GED you will often not be asked to solve an inequality — you will be asked to write one. A question describes a situation in words ("Amir can spend no more than $60 on tickets that cost $12 each") and asks: Which inequality represents this situation?

In this lesson you will learn to:

  • translate key phrases like "at least", "no more than" and "exceeds" into the correct symbol,
  • define the variable clearly,
  • build the left side from a fixed amount and a rate,
  • handle tricky wording such as "5 less than a number" versus "5 is less than a number",
  • write inequalities from geometry and percent situations.

Why it matters on the GED: "Which inequality represents…" is one of the most common algebra item types. The wrong answer choices are built from predictable mistakes — the wrong symbol, the numbers swapped, the fixed fee forgotten — so if you know those traps, these questions become fast points.

Words to know

Key words

  • Variable — the letter for the unknown quantity; always say what it means (example: "let t = number of tickets").
  • Expression — numbers and variables without a comparison sign (example: 12t + 5).
  • Inequality — two expressions joined by <, >, \le or \ge (example: 12t + 5 \le 60).
  • Rate — an amount per unit, which multiplies the variable (example: $12 per ticket → 12t).
  • Fixed amount — a one-time amount that is added or subtracted once (example: a $5 service fee → +5).
  • Limit — the number on the other side of the inequality (a budget, a capacity, a goal).
  • Inclusive — the limit itself is allowed (\le or \ge), as in "at most 60".
  • Strict — the limit itself is not allowed (< or >), as in "less than 60".
  • Threshold — a starting point after which a charge begins (example: $0.05 per text over 500 → 0.05(t - 500)).

All Math words in the glossary

Translate the words into a symbol

Memorize this table. It is the heart of writing inequalities.

Words in the problemSymbolExampleInequality
less than, fewer than, below<fewer than 30 studentss < 30
more than, greater than, exceeds, above>a speed above 55 mphv > 55
at most, no more than, maximum, not exceed, up to\leno more than $60c \le 60
at least, no less than, minimum, not below\geat least 18 years olda \ge 18
Four number lines for the translation table. fewer than 30: open circle at 30 shaded left (s < 30). above 55: open circle at 55 shaded right (v > 55). no more than 60: closed circle at 60 shaded left (c <= 60). at least 18: closed circle at 18 shaded right (a >= 18). fewer than 30 students: s < 30 25 26 27 28 29 30 31 32 33 34 35 a speed above 55 mph: v > 55 50 51 52 53 54 55 56 57 58 59 60 no more than $60: c ≤ 60 55 56 57 58 59 60 61 62 63 64 65 at least 18 years old: a ≥ 18 13 14 15 16 17 18 19 20 21 22 23

Two helpful ideas:

  • "No more than" = "at most" = \le. If you can't have more, you can have that amount or less.
  • "No less than" = "at least" = \ge. If you can't have less, you can have that amount or more.

Inclusive or strict? Ask: Is the limit itself allowed? A bridge with a 10-ton limit allows exactly 10 tons (w \le 10). A ride for children taller than 48 inches does not allow exactly 48 inches (h > 48).

Careful with "less than".

  • "5 less than a number" is an expression: x - 5 (start with x, take 5 away — the order reverses).
  • "5 is less than a number" is an inequality: 5 < x.

The little word "is" tells you where the inequality sign goes.

Example A. "Twice a number, increased by 7, is at least 25." → 2x + 7 \ge 25.

Tape model: two blocks labelled x (twice a number, 2x) and a block labelled 7 placed end to end, compared with a block labelled 25 using the at-least sign. The inequality is 2x + 7 >= 25. “Twice a number, increased by 7, is at least 25” x x 7 twice a number: 2x ≥ 25 2x + 7 ≥ 25

Example B. "Three less than a number is no more than 10." → x - 3 \le 10.

Build the expression: fixed amount + rate × variable

Most word problems have the same structure:

\underbrace{\text{fixed amount}}_{\text{once}} + \underbrace{\text{rate} \times \text{variable}}_{\text{per item}} \;\;(\text{symbol})\;\; \text{limit}

How to tell which number is which:

  • The rate comes with "per", "each", "every", "an hour", "a mile". It multiplies the variable.
  • The fixed amount happens once: a fee, a starting balance, a person's own weight, money already saved. It is added (or subtracted, for costs in a profit problem).

Example C. A plumber charges $75 for a house call plus $60 per hour. The customer will pay no more than $300.

  • Variable: h = hours worked.
  • Rate: $60 per hour → 60h. Fixed: $75 → +75.
  • "No more than" → \le.
75 + 60h \le 300
Bar model of the plumber's bill: a 75 dollars house-call block that is paid once, followed by 60 dollars blocks, one per hour (three hours shown, labelled 60h), compared with a limit line at 300 dollars. The inequality is 75 + 60h <= 300. 75 + 60h ≤ 300 (shown with h = 3) $75 call no more than $300 fixed: once $60 per hour × h hours: 60h

Example D. Sam has $25 and saves $15 a week. He needs at least $200.

25 + 15w \ge 200

Example E (profit). A club sells cookies at $1.50 each and spent $48 on supplies. Profit must be at least $120.

1.50c - 48 \ge 120

Profit is money in minus costs, so the fixed cost is subtracted.

Harder set-ups: thresholds, geometry and several parts

Thresholds ("over", "after the first"). A plan charges $30 a month plus $0.05 for each text over 500. If t is the number of texts (with t > 500), only t - 500 texts are charged:

30 + 0.05(t - 500) \le 45
Bar of 800 texts split at 500: the first 500 texts are included in the 30 dollars monthly fee; only the t - 500 = 300 texts over 500 are charged at 0.05 dollars each, which is 15 dollars. The bill is 30 dollars + 15 dollars = 45 dollars. 30 + 0.05(t − 500) ≤ 45 (shown with t = 800) first 500 texts: included in $30 t − 500 = 300 300 × $0.05 = $15 0 500 800 bill: $30 + $15 = $45

Geometry. Use a formula from the formula sheet, then attach the limit. A rectangle has width w and length 2w + 3. Its perimeter must be less than 54 ft:

P = 2l + 2w = 2(2w + 3) + 2w = 6w + 6 \quad\Rightarrow\quad 6w + 6 < 54
Rectangle with length 2w + 3 on the top and bottom and width w on the left and right sides, right angles marked. Perimeter P = 2(2w + 3) + 2w = 6w + 6, which must be less than 54 ft. Length = 2w + 3, width = w 2w + 3 2w + 3 w w P = 2(2w + 3) + 2w = 6w + 6 6w + 6 < 54

Several income parts. A club already earned $80 from brownies (40 at $2), will sell cookies at $1.50 each, and spent $48 on supplies. Profit must be at least $200:

1.50c + 80 - 48 \ge 200

Writing it in steps (works every time):

  1. Read the question twice and name the variable — what are you counting?
  2. Write what changes with the variable (rate × variable).
  3. Add or subtract what happens once.
  4. Find the limit and the limit words; choose the symbol.
  5. Test your inequality with a simple number to see if it "behaves" like the story.
Formula

Formulas you may need

The general pattern is not on the formula sheet — memorize it:

\text{fixed amount} + (\text{rate})x \;\;\square\;\; \text{limit}, \qquad \square \in \{<,\ >,\ \le,\ \ge\}
  • x = the quantity you don't know (hours, items, miles…).
  • rate = amount per one unit of x.
  • fixed amount = one-time amount (can be 0).

Formulas that are on the GED formula sheet and often appear inside inequalities:

\text{Perimeter of a rectangle: } P = 2l + 2w \qquad \text{Total cost} = (\text{number of units}) \times (\text{price per unit})
Worked example

Worked example 1 (easy): one phrase

Problem: A job requires applicants to be at least 21 years old. Write an inequality for an allowed age a.

  1. "At least" means 21 or more, so the limit is included.
  2. The symbol is \ge.

Answer: a \ge 21

Number line from 16 to 26. Closed circle at 21 with shading and an arrow to the right: the graph of a >= 21. Test value 21 is marked true (in the solution set). Test value 20 is marked false (not a solution). at least 21 years old: a ≥ 21 16 17 18 19 20 21 22 23 24 25 26 years 21 ✓ 20 ✗

Check: A 21-year-old qualifies: 21 \ge 21 ✓. A 20-year-old doesn't: 20 \ge 21 is false ✓.

Worked example

Worked example 2: a rate and a budget

Problem: Concert tickets cost $45 each, plus a one-time $8 processing fee per order. Leah can spend no more than $200. Write an inequality for the number of tickets t.

  1. Rate: $45 per ticket → 45t.
  2. Fixed amount: $8 once → +8.
  3. "No more than" → \le; limit 200.

Answer: 45t + 8 \le 200

Bar model of Leah's order: an 8 dollars processing fee block plus four 45 dollars ticket blocks, 180 dollars, for a total of 188 dollars, under the 200 dollars budget line. A dashed red fifth ticket brings the total to 233 dollars, over budget. The inequality is 45t + 8 <= 200. 45t + 8 ≤ 200 budget $200 $8 4 tickets × $45 = $180; total $188 ✓ 5 tickets: $233 ✗ $0

Check: Try t = 4: 180 + 8 = 188 \le 200 ✓ (she can afford 4). Try t = 5: 225 + 8 = 233, over budget ✓ — the inequality matches the story.

Worked example

Worked example 3 (real world): a weight limit

Problem: A van can carry at most 1,200 lb. The driver weighs 190 lb. Cases of water weigh 42 lb each. Write an inequality for the number of cases c.

  1. Rate: 42 lb per case → 42c.
  2. Fixed amount: the driver, 190 lb → +190.
  3. "At most" → \le 1200.

Answer: 190 + 42c \le 1200

Bar model of the van load: a 190 lb block for the driver plus 24 cases of 42 lb each, 1,008 lb, for a total of 1,198 lb, just under the 1,200 lb limit line. A dashed red 25th case brings the total to 1,240 lb, too heavy. The inequality is 190 + 42c <= 1200. 190 + 42c ≤ 1200 190 lb at most 1,200 lb driver 24 cases × 42 lb = 1,008 lb total 1,198 lb ✓ 25 cases: 1,240 lb ✗

Check: With 24 cases: 190 + 1008 = 1198 \le 1200 ✓. With 25: 190 + 1050 = 1240 — too heavy ✓.

Worked example

Worked example 4 (GED-hard): perimeter limit

Problem: A garden is a rectangle whose length is 3 ft more than twice its width w. The fencing available is less than 54 ft. Write an inequality for w.

  1. Length: "3 more than twice the width" → l = 2w + 3.
  2. Perimeter (formula sheet): P = 2l + 2w = 2(2w + 3) + 2w.
  3. Simplify: 4w + 6 + 2w = 6w + 6.
  4. "Less than 54" → <.

Answer: 6w + 6 < 54 (which solves to w < 8).

Garden rectangle drawn to scale with width 7 ft and length 17 ft (3 more than twice the width), right angles marked. Perimeter 2(17) + 2(7) = 48 ft, which is less than 54 ft. In general 6w + 6 < 54. Example: w = 7 ft, l = 2(7) + 3 = 17 ft 17 ft 17 ft 7 ft 7 ft P = 2(17) + 2(7) = 48 < 54 ✓ 6w + 6 < 54

Check: If w = 7: l = 17, P = 34 + 14 = 48 < 54 ✓. If w = 8: l = 19, P = 38 + 16 = 54, not less than 54 ✓.

Worked example

Worked example 5 (work backwards): write, then solve

Problem: A school club sells cookies for $1.50 each. It already sold 40 brownies at $2 each and spent $48 on supplies. The club wants a total profit of at least $200. What is the least number of cookies it must sell?

  1. Variable: c = cookies sold.
  2. Money in: 1.50c + 40(2) = 1.50c + 80. Money out: 48.
  3. Inequality: 1.50c + 80 - 48 \ge 200.
  4. Simplify: 1.50c + 32 \ge 200.
  5. Subtract 32: 1.50c \ge 168.
  6. Divide by 1.50: c \ge 112.

Answer: 112 cookies

Two bars. Money in: brownies 80 dollars plus 112 cookies at 1.50 dollars, 168 dollars, for 248 dollars. Money out: 48 dollars for supplies, lined up under the end of the money-in bar. What is left, the profit, is 248 - 48 = 200 dollars, which meets the goal of at least 200 dollars. 1.50c + 80 − 48 ≥ 200 (c = 112) money in brownies $80 112 × $1.50 = $168 money out $48 supplies → profit = 248 − 48 = $200 ≥ $200 goal ✓

Check: 1.50(112) + 80 - 48 = 168 + 32 = 200 \ge 200 ✓. With 111 cookies: 166.50 + 32 = 198.50, short ✓.

Calculator tip

Calculator tip (TI-30XS MultiView)

Writing an inequality is a reading skill — the calculator can't choose the symbol or decide which number is the rate. Many "which inequality represents…" items are quick to do without it.

Use the calculator to test a choice with a sample number. For 190 + 42c \le 1200 with c = 24: type 190 + 42 × 24 enter → 1198. Compare with 1,200 yourself.

  • Use (−) for negative values and parentheses for thresholds: 30 + 0.05 × ( 800 − 500 ) enter → 45.
  • Use n/d for fractional rates and toggle to see them as decimals.
Common mistake

Common mistakes

  • "No more than" turned into \ge. ✗ 12t \ge 60. ✓ "No more than $60" means $60 or less: 12t \le 60.
  • Strict vs. inclusive mixed up. "At most 1,200 lb" written as < 1200. ✗ ✓ "At most" includes 1,200: \le.
  • Rate and fixed amount swapped. $75 call fee + $60/hour written as 60 + 75h. ✗ ✓ The number with "per hour" multiplies h: 75 + 60h.
  • Adding instead of multiplying the rate. t + 12 \le 60 ✗. ✓ "$12 each" means 12t.
  • Forgetting the fixed amount. 15w \ge 200 when Sam already has $25. ✗ ✓ 25 + 15w \ge 200.
  • Reversing "less than" expressions. "5 less than x" written as 5 - x. ✗ ✓ x - 5.
Test strategy

GED test strategy

  • Check the symbol first. Find the limit words and decide the symbol. Cross off every choice with the wrong symbol — often two choices go immediately.
  • Then check the structure. Which number has "per/each"? It must be next to the variable.
  • Plug in a friendly number. Choose a value for the variable (like 1 or 10), compute the real-life total from the story, and see which inequality gives that same total on its left side.
  • Watch the direction of subtraction in profit and threshold problems: money in − money out; texts − 500.
  • Read twice: is the question asking you to write the inequality or to solve it? Don't solve if they only want the inequality.
  • Units: make sure both sides are in the same units (dollars with dollars, feet with feet).

Check yourself

Write inequalities from contextNo calculator

Which inequality represents the statement “a number x is at least 18”?

Answer choices
Hint

Does “at least 18” allow the number to be exactly 18? Can it be bigger?

Check yourself

Write inequalities from contextNo calculator

Movie tickets cost $12 each. Amir can spend no more than $60 on tickets. Which inequality represents the number of tickets t he can buy?

Answer choices
Hint

How much do t tickets cost? Then translate “no more than”.

Summary

  • "At most / no more than" → \le; "at least / no less than" → \ge; "more than / exceeds" → >; "less than / fewer than" → <.
  • Name the variable, then write fixed amount + rate × variable, then the symbol and the limit.
  • "Is" marks where the inequality sign goes: "5 less than x" is x - 5, but "5 is less than x" is 5 < x.
  • Thresholds use (t - 500); geometry uses the formula sheet (P = 2l + 2w).
  • Test your inequality with a sample number to be sure it matches the story.
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