Lesson 8.4Write and Represent Inequalities
Start hereWords, worked example, and sentence starters
Learning target I can write inequalities to represent real-world situations.
Start hereWords for this lesson
Read each word before you begin. Say it out loud.
| Word | What it means | Example |
|---|---|---|
| InequalitySpanish: Desigualdad | A math sentence that compares two sides with <, >, ≤, or ≥. | x ≥ 18 |
| Greater thanSpanish: Mayor que | The > sign, showing one number is bigger. | 8 > 3 — eight is larger than three |
| Less thanSpanish: Menor que | The < sign, showing one number is smaller. | 2 < 7 — two is smaller than seven |
| At least / At mostSpanish: Al menos / A lo más | 'At least' means ≥. 'At most' means ≤. | 'At least 10 players' → p ≥ 10 (could be 10, 11, 12, ...); 'At most 5 tries' → t ≤ 5 (could be 5, 4, 3, ...) |
| VariableSpanish: Variable | A letter that stands for a number that is unknown or can change. | the x in x ≥ 18 |
| No more thanSpanish: No más de | A phrase meaning less than or equal to (≤). | “No more than 8 people” means people ≤ 8. |
How it worksWorked examplewriting an inequality for a constraint
These numbers are not on your problems. The steps are. Follow them with your own numbers.
A club trip requires each member to save at least $45. Write an inequality for the amounts that work.
- Name the variable: let s = the dollars a member saves.
- Find the key phrase: 'at least' means the amount can equal 45 or be more.
- That phrase matches the symbol ≥, so write s ≥ 45.
- Test values: 45 works and 52 works, but 39 falls below the limit.
Answer: s ≥ 45, so any amount of 45 dollars or more is a solution.
Now you trySame steps, your turn
The steps are the same as the worked example. The numbers are yours. Do the work.
A team must win at least 7 games to reach the playoffs. Write an inequality for the wins.
- Name the variable: let w = .
- The phrase 'at least' means w can be equal to or it.
- Write the inequality: w 7
- Test: is 6 a solution? Is 11 a solution?
Say it and write itSentence starters
Finish each sentence out loud with a partner. Then use them in your writing.
- The phrase tells me the symbol is .
- My inequality is because .
- The value (is / is not) a solution because .
- This inequality has many solutions because .
Word bank inequalityat leastat mostmore thanless thangreater thansymbolconstraintvariablesolutionboundaryrange
Watch outA common mistake
A common mistake in Write Inequalities is using a strict symbol (> or <) for a phrase that actually includes the boundary value. For 'the suspect is at least 18 years old,' students often write a > 18 instead of the correct a ≥ 18 — this wrongly excludes 18 itself, even though 'at least 18' means 18 counts as a match. The same error happens in reverse with 'at most': for 'the getaway bag holds at most 12 items,' students write n < 12 instead of n ≤ 12. Always ask: does the key phrase include the number ('at least,' 'at most,' 'no more than,' 'no fewer than' → use ≥ or ≤) or exclude it ('more than,' 'fewer than' → use > or <)?
Lesson 8.4Write and Represent Inequalities
Version ASupported practice
Learning target I can write inequalities to represent real-world situations.
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1Mark and label each value on the number line.
Place markers on the number line for each value, then decide: is 5 > 3 true or false? Is 2 ≥ 7 true or false?
Hint On a number line, greater numbers are to the right. 5 is to the right of 3, so 5 > 3 is true.
Show your thinking
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2Circle the letter of the best answer. Show how you know.
Which inequality represents: 'The temperature is less than 40 degrees'?
- At < 40
- Bt > 40
- Ct ≤ 40
- Dt = 40
Hint Find the comparison phrase: 'less than 40 degrees'. Does 40 itself count, or only temperatures below it?
Show your work -
3Circle the letter of the best answer. Show how you know.
Which inequality represents: 'You must be at least 48 inches tall to ride'?
- Ah ≥ 48
- Bh > 48
- Ch < 48
- Dh ≤ 48
Hint 'At least 48' — ask yourself: can a rider who is EXACTLY 48 inches tall ride? That decides whether 48 is included.
Show your work -
4Circle the letter of the best answer. Show how you know.
Which inequality represents: 'A store can hold no more than 75 people'?
- Ap ≤ 75
- Bp < 75
- Cp ≥ 75
- Dp > 75
Hint 'No more than 75' — could the store hold exactly 75 people? Could it hold 76?
Show your work -
5Circle the letter of the best answer. Show how you know.
Which inequality represents: 'The speed limit is more than 25 mph'?
- As > 25
- Bs ≥ 25
- Cs < 25
- Ds = 25
Hint 'More than 25' — is exactly 25 included, or only speeds above 25?
Show your work
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6Find the mistake. Explain it, then show the correct work.
Spot the Detective's Mistake
- 1ClueThe witness says the suspect is at least 16 years old.
- 2Choose a variableLet a stand for the suspect's age.
- 3Write the inequalitya > 16
Which step has the mistake? Explain what went wrong, then show the correct work.
Hint Does 'at least 16' allow the suspect's age to be exactly 16, or must it be older?
Correct workCorrect answer:
Explain your thinkingHow do the words 'at least' and 'at most' change the inequality symbol compared to 'more than' and 'fewer than'?
Sentence starter 'At least' uses ≥ instead of > because it means the value ___ be equal to the number, while 'more than' means it must be ___.
Lesson 8.4Write and Represent Inequalities
Version BCore practice
Learning target I can write inequalities to represent real-world situations.
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1Write the letter of the matching item on each line.
Match each situation to its inequality
- Item 1
- Item 2
- Item 3
- Item 4
- A
- B
- C
- D
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2Complete the table. Show how you found each value.
Identify the key phrase, symbol, and write the inequality for each situation.
Situation Key Phrase Symbol Inequality A package weighs less than 50 pounds. You need at least $25 to buy the game. No more than 30 students can join the club. Show your work -
3Write the name of the correct group on each line.
Test x = 10 in each inequality. Sort by whether it works.
- x ≥ 10
- x ≤ 15
- x > 8
- x > 10
- x < 10
- x ≤ 5
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4Circle the letter of the best answer. Show how you know.
A parking garage charges if your stay is more than 2 hours. Which inequality represents times t that are charged?
- At > 2
- Bt ≥ 2
- Ct < 2
- Dt ≤ 2
Show your work
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5Find the mistake. Explain it, then show the correct work.
Crack the Inequality Error
- 1ClueThe report says no fewer than 9 witnesses saw the suspect.
- 2Choose a variableLet w stand for the number of witnesses.
- 3Write the inequalityw < 9
Which step has the mistake? Explain what went wrong, then show the correct work.
Correct workCorrect answer:
Explain your thinkingHow do the words 'at least' and 'at most' change the inequality symbol compared to 'more than' and 'fewer than'?
Lesson 8.4Write and Represent Inequalities
ChallengeExtension
Learning target I can write inequalities to represent real-world situations.
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1Complete the table. Show how you found each value.
For each situation, decide if it needs an equation or inequality and explain why.
Equation or Inequality? Model Why? A detective has exactly 12 cases. A detective has more than 12 cases. A detective has at most 12 cases. Show your work
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2Find the mistake. Explain it, then show the correct work.
Find the Inequality Error
- 1SituationNo more than 25 students can join the field trip
- 2Identify key phrase'No more than' means greater than
- 3Write inequalitys > 25
Which step has the mistake? Explain what went wrong, then show the correct work.
Correct workCorrect answer: -
3Answer in complete sentences.
Write two different real-world situations: one that uses > and one that uses ≥. Explain why the situations need different symbols even though both mean 'bigger.'
Write your answer