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➗ Math · Numbers and Algebra · Lesson 8 of 8 · 7 min

Expressions, inequalities, word problems, estimation and rounding

📝 The hardest part of a word problem is turning sentences into math. Learn the translation words and the rest is arithmetic.

💡 Key idea

Translate the words carefully, and flip the inequality sign whenever you multiply or divide by a negative.

🧠 The idea

Translate words into an EXPRESSION: "more than" and "increased by" mean add, "less than" means subtract in reverse order (5 less than x is x − 5), "of" and "times" mean multiply, and "per" and "quotient" mean divide. An INEQUALITY uses <, >, ≤ (at most) or ≥ (at least). Solve it like an equation, with one exception: when you multiply or divide both sides by a negative number, FLIP the inequality sign. From −2x > 8 you get x < −4. For multistep word problems, write down what you know, decide what you are finding, and solve one step at a time. ROUNDING: look at the digit to the right of the place you are rounding to; 5 or more rounds up, 4 or less leaves it. 3.46 rounded to the nearest tenth is 3.5. ESTIMATION means rounding before you compute to check that an answer is reasonable: 49 × 21 is about 50 × 20 = 1,000.

✏️ Worked example

A phone plan costs $25 a month plus $0.10 per text. Dana wants to spend at most $40 this month. How many texts can she send?
Step 1: Let t = number of texts. Cost = 25 + 0.10t.
Step 2: "At most" means ≤: 25 + 0.10t ≤ 40.
Step 3: Subtract 25 from both sides: 0.10t ≤ 15.
Step 4: Divide both sides by 0.10 (positive, so no flip): t ≤ 150.
Answer: at most 150 texts.

📌 Takeaways

  • "Less than" reverses the order: 5 less than x is x − 5.
  • Flip the inequality sign when multiplying or dividing by a negative.
  • Round to the place asked; estimate first to catch silly answers.

📖 Terms in this lesson

Inequality: A statement that one quantity is greater or less than another, using <, >, ≤ or ≥; for example, x + 2 > 5 means x > 3.

✅ Check yourself

Question 1 of 3

Which expression means "8 less than three times a number n"?

  1. 8 − 3n
  2. 3(n − 8)
  3. 3n − 8
  4. 3n + 8
Show the answer and rationale

Answer: C

Three times a number is 3n, and 8 less than that is 3n − 8. The expression 8 − 3n reverses the order, a common trap with "less than", and 3(n − 8) subtracts before multiplying.

Question 2 of 3

Solve −3x + 4 ≤ 19.

  1. x ≥ −5
  2. x ≤ 5
  3. x ≥ 5
  4. x ≤ −5
Show the answer and rationale

Answer: A

Subtract 4: −3x ≤ 15. Divide by −3 and flip the sign: x ≥ −5. The answer x ≤ −5 forgets to flip the sign, and the answers with positive 5 lose the negative when dividing.

Question 3 of 3

What is 47.862 rounded to the nearest hundredth?

  1. 47.9
  2. 48
  3. 47.86
  4. 47.87
Show the answer and rationale

Answer: C

The hundredths digit is 6, and the digit to its right is 2, which is less than 5, so the 6 stays: 47.86. The answer 47.87 rounds up when it should not, and 47.9 rounds to the nearest tenth instead.

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