How to turn words into expressions, equations and inequalities
TEAS 7 outline: Model real-world situations with expressions, equations and inequalities
Also covers HESI A2 Math (Basic algebra)
- Equations use =; inequalities use <, >, ≤ or ≥; expressions use neither.
- "Less than" flips the order: "5 less than x" is x − 5.
- "At least" means ≥ and "at most" means ≤; both include the boundary.
- Open circle for < or >; closed circle for ≤ or ≥.
- Multiply or divide an inequality by a negative, and the sign flips.
Key facts
- An expression has no equal or inequality sign (12s + 30); an equation uses = (12s + 30 = 246); an inequality uses <, >, ≤ or ≥.
- "Less than" and "subtracted from" flip the order: "5 less than x" is x − 5.
- Parentheses follow the words: "3 times the sum of x and 4" is 3(x + 4), but "the sum of 3 times x and 4" is 3x + 4.
Show all 7 key factsShow fewer
- "At least" and "no less than" mean ≥; "at most", "no more than" and "does not exceed" mean ≤.
- "More than", "exceeds" and "over" mean >; "less than", "fewer than" and "under" mean <.
- On a number line, an open circle means the boundary is not included (< or >) and a closed circle means it is included (≤ or ≥); shade right for greater, left for less.
- Multiplying or dividing both sides of an inequality by a negative number flips the sign: −3x > 12 becomes x < −4.
Quick check1 question · 2 more at the end
A sign at a hospital parking garage says that a vehicle's height can be no more than 7 feet. Which inequality shows the allowed heights, h, in feet?
Correct answer: C.
"No more than 7" means the height can be 7 feet or anything below it, so the boundary 7 is allowed and the sign is ≤: h ≤ 7. "At most" and "no more than" both mean ≤; "at least" and "no less than" both mean ≥. If the boundary number itself is allowed, the inequality needs the "or equal to" line.
A This leaves out 7 itself, but a vehicle exactly 7 feet tall is allowed, so the sign needs the "or equal to" part.
B This reverses the meaning: ≥ is "at least 7," which would allow only vehicles 7 feet or taller.
C Correct: "no more than 7" means 7 or less, which is h ≤ 7.
D This both reverses the direction and leaves out 7; it describes vehicles taller than the limit.
Full lesson
Each part starts with its main idea. Open a part when you want the details.
The short answer: Name the unknown with a letter, then translate piece by piece: more than means add, less than means subtract in reversed order, a per-item rate multiplies the count, and is means equals. At least means ≥ and at most means ≤. On a number line, < and > get an open circle, while ≤ and ≥ get a closed circle.
1 What is the difference between an expression, an equation and an inequality?An expression is like a phrase, while equations and inequalities are full sentences. 1 min
An expression is like a phrase, while equations and inequalities are full sentences. An expression joins numbers and a variable (a letter for the unknown) with operations. An equation sets two expressions equal, and an inequality compares them.
| Type | Uses | Example |
|---|---|---|
| Expression | No sign in the middle | 12s + 30 |
| Equation | = | 12s + 30 = 246 |
| Inequality | <, >, ≤ or ≥ | 12s + 30 ≤ 246 |
2 How do I translate words into math?Most word problems are built from the same small set of signal words. 1 min
Most word problems are built from the same small set of signal words.
| Words | Symbol |
|---|---|
| sum, more than, increased by, plus | + |
| difference, less than, decreased by, subtracted from | − |
| product, times, twice (2 ×), a rate "per" or "for each" item | × |
| quotient, divided by, split equally | ÷ |
| is, equals, totals | = |
The one rule to remember: "less than" and "subtracted from" flip the order. "5 less than x" is x − 5, not 5 − x.
Parentheses follow the words. "3 times the sum of x and 4" is 3(x + 4), but "the sum of 3 times x and 4" is 3x + 4.
Example 1. A clinic pays a courier $30 a month plus $12 for each lab sample picked up. Last month's bill was $246. How many samples, s, were picked up?
- Name the unknown: s = number of samples.
- The changing part is $12 per sample, so 12s.
- The fixed part is $30.
- Write the equation: 12s + 30 = 246.
- Solve: subtract 30 to get 12s = 216, then divide by 12, so s = 18.
- Check: 12 × 18 + 30 = 246.
Classic wrong turn: writing 30s + 12. The variable always goes with the per-item amount, never the one-time fee.
3 Which words turn a sentence into an inequality?Inequality words tell you whether the boundary number itself is allowed. 1 min
Inequality words tell you whether the boundary number itself is allowed.
| Words | Symbol | Boundary allowed? |
|---|---|---|
| at least, no less than, minimum | ≥ | Yes |
| at most, no more than, maximum, does not exceed | ≤ | Yes |
| more than, greater than, exceeds, over | > | No |
| less than, fewer than, under | < | No |
Watch for the word "is." "x is less than 5" compares (x < 5), but "5 less than x" subtracts (x − 5).
Example 2. A home health aide earns $22 an hour and wants to earn at least $500 this week. How many whole hours, h, must she work?
- Earnings are 22h.
- "At least $500" allows exactly $500, so use ≥: 22h ≥ 500.
- Divide both sides by 22: h ≥ 22.7 (rounded).
- Test whole numbers: 22 hours pays $484 (too little), and 23 hours pays $506. She needs 23 hours.
Classic wrong turn: rounding 22.7 down to 22. Always test the whole number in the inequality. "At least" goals usually round up, and "at most" budgets usually round down.
One solving rule: multiplying or dividing both sides by a negative number flips the sign, so −3x > 12 becomes x < −4.
4 How do I graph an inequality on a number line?The circle shows whether the boundary counts, and the shading shows every other answer. 1 min
The circle shows whether the boundary counts, and the shading shows every other answer.
- Put the variable on the left (rewrite 7 > x as x < 7).
- Use an open circle for < or >, because the boundary is not a solution. Use a closed (filled) circle for ≤ or ≥.
- Shade right for > or ≥ (bigger numbers) and left for < or ≤.
Some books use a parenthesis "(" for an open circle and a bracket "[" for a closed one.
Example 3. A hospital playroom is for children under 6 years old. Graph the allowed ages, a.
- "Under" means less than, and 6 is not allowed: a < 6.
- Draw an open circle at 6.
- Shade to the left, toward smaller numbers.
Classic wrong turn: filling in the circle at 6. A child who just turned 6 is not "under 6."
Common mistakes
The traps students fall into most. Read these before you practice.
Watch out: "7 less than n" is n − 7, not 7 − n.
Watch out: "At least 500" is ≥ 500, not > 500: the boundary number itself counts.
Watch out: In a "fee plus a rate" problem, the variable goes with the per-item rate (12s + 30), not with the one-time fee (30s + 12).
Watch out: Round a whole-number answer by testing it, not by habit: an "at least" goal of h ≥ 22.7 needs 23 hours, not 22.
Watch out: x < 6 gets an open circle at 6, not a closed one, because 6 is not part of the answer.
How to remember it
Think of "at least" as a floor and "at most" as a ceiling. You can stand on a floor and touch a ceiling, so both include the boundary number (≥ or ≤, with a closed circle).
Self-check:
- Translate "3 less than 4 times a number is 25" into an equation and solve it.
- Write "no more than 1,200 mL" as an inequality using m.
- Say whether x > −2 gets an open or closed circle, and which way to shade.
Show answers
- 4n − 3 = 25, so n = 7.
- "4 times a number" is 4n. "3 less than" that flips the order: 4n − 3.
- "Is 25" means = 25.
- Add 3 to both sides: 4n = 28.
- Divide both sides by 4: n = 7.
- Check: 4 × 7 − 3 = 28 − 3 = 25.
- m ≤ 1,200.
- "No more than" allows exactly 1,200 mL, so use ≤.
- Open circle at −2, shade right.
- The sign is >, so −2 itself is not a solution. Use an open circle.
- "Greater than" means bigger numbers, so shade to the right.