Lesson 8.6Solve and Graph Inequalities
Start hereWords, worked example, and sentence starters
Learning target I can solve an inequality and graph its solution set on a number line.
Start hereWords for this lesson
Read each word before you begin. Say it out loud.
| Word | What it means | Example |
|---|---|---|
| SolveSpanish: Resolver | To find the number that makes it true. | x + 3 > 10 → subtract 3 → x > 7 — any number greater than 7 works |
| GraphSpanish: Graficar | To show answers on a number line with circles and shading. | x > 7: open circle at 7, shade right — shows all solutions at a glance |
| SolutionSpanish: Solución | A number that makes the equation or inequality true. | x = 8 is a solution to x > 7 because 8 > 7 is true; x = 5 is not because 5 > 7 is false |
| SubstituteSpanish: Sustituir | To put a number in place of a letter. | Substitute x = 5 into x + 3 > 10: 5 + 3 = 8, and 8 > 10 is false, so x = 5 is not a solution |
| Solution setSpanish: Conjunto solución | All the numbers that make the inequality true. | For x > 4, the solution set is every number greater than 4, such as 5, 6, 7, ... |
| Inverse operationSpanish: Operación inversa | Two math actions that undo each other, like × and ÷. | To solve x + 3 > 7, use the inverse of adding 3: subtract 3 to get x > 4. |
How it worksWorked example
These numbers are not on your problems. The steps are. Follow them with your own numbers.
Solve x + 3 > 53. The inverse of adding 3 is subtracting 3.
- I subtract 3 from both sides: x + 3 − 3 > 53 − 3, so x > 50.
- The symbol > does not include 50, so I use an open circle at 50.
- 'Greater than' shades right. Check: x = 60 gives 60 + 3 = 63 > 53. True!
Now you trySame steps, your turn
The steps are the same as the worked example. The numbers are yours. Do the work.
Solve x − 5 ≤ 8. What is the inverse of subtracting 5?
Say it and write itSentence starters
Finish each sentence out loud with a partner. Then use them in your writing.
- Solving an inequality is like solving an equation because I still use , but it is different because the answer is instead of .
Word bank SolveGraphSolutionSubstituteSolution setInverse operationinverse operationsmany valuesone valuegraphsolution setnumber line
Watch outA common mistake
A common mistake in Solve and Graph Inequalities is shading the number line in the wrong direction after solving. For example, a student solves x + 4 > 10 correctly to get x > 6, but then shades LEFT toward smaller numbers instead of RIGHT toward greater numbers. The shading direction always matches the solved symbol, not the original problem: '>' and '≥' shade right (greater numbers), '<' and '≤' shade left (smaller numbers) — check this AFTER you solve, not before. To catch this mistake, substitute one shaded value and one unshaded value back into the original inequality: the shaded value should make it true, and the unshaded value should make it false.
Lesson 8.6Solve and Graph Inequalities
Version ASupported practice
Learning target I can solve an inequality and graph its solution set on a number line.
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1Mark and label each value on the number line.
Solve x + 3 > 9, then graph the solution on the number line.
Hint Subtract 3 from both sides: x > 6.
Show your thinking
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2Circle the letter of the best answer. Show how you know.
Solve: x + 6 > 14
- Ax > 8
- Bx < 8
- Cx = 8
- Dx > 20
Hint Solve it like an equation: x has 6 added to it. The > symbol just rides along.
Show your work -
3Circle the letter of the best answer. Show how you know.
Solve: x − 9 ≤ 3
- Ax ≤ 6
- Bx ≤ 12
- Cx ≥ 12
- Dx = 12
Hint x has 9 subtracted from it. Undo that first; the ≤ symbol keeps its direction.
Show your work -
4Circle the letter of the best answer. Show how you know.
Solve: x + 10 ≥ 25
- Ax ≥ 15
- Bx ≤ 15
- Cx = 15
- Dx ≥ 35
Hint What's being done to x? 10 is added. Plan the inverse operation before you look at the choices.
Show your work -
5Circle the letter of the best answer. Show how you know.
Solve: x − 4 < 11
- Ax < 7
- Bx < 15
- Cx > 15
- Dx = 15
Hint x has 4 taken away. Which operation puts that 4 back?
Show your work
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6Find the mistake. Explain it, then show the correct work.
Catch the Wrong Inverse Operation
- 1Inequalityx + 6 > 14
- 2Inverse operationAdd 6 to both sides
- 3Simplifyx > 20
- 4GraphOpen circle at 20, shade right
Which step has the mistake? Explain what went wrong, then show the correct work.
Hint The inequality has + 6. To undo adding 6, do you add or subtract?
Correct workCorrect answer:
Explain your thinkingHow is solving an inequality similar to and different from solving an equation?
Sentence starter Solving an inequality is like solving an equation because I still use ___, but it is different because the answer is ___ instead of ___.
Lesson 8.6Solve and Graph Inequalities
Version BCore practice
Learning target I can solve an inequality and graph its solution set on a number line.
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1Complete the table. Show how you found each value.
Solve each inequality, describe the inverse operation, and describe the graph.
Inequality Inverse Operation Solution Graph Description x + 11 ≥ 20 x − 3 < 7 x + 8 ≤ 20 Show your work -
2Write the name of the correct group on each line.
Solve each inequality, then test if x = 10 is a solution.
- x + 2 > 8 (x > 6)
- x − 5 ≥ 3 (x ≥ 8)
- x + 1 ≤ 15 (x ≤ 14)
- x + 3 > 15 (x > 12)
- x − 2 ≥ 9 (x ≥ 11)
- x + 6 < 10 (x < 4)
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3Decide whether each pair balances. Check Yes or No, then explain.
Solve the inequality x + 7 ≥ 19 using the balance scale. What is the boundary value?
Left side Right side Balanced? Explain your choice
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4Circle the letter of the best answer. Show how you know.
Is x = 5 a solution to x + 4 > 12?
- ANo, because 5 + 4 = 9, and 9 is not greater than 12
- BYes, because 5 + 4 = 9
- CYes, because 5 > 4
- DNo, because 5 < 12
Show your work
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5Find the mistake. Explain it, then show the correct work.
Catch the Open vs. Closed Circle Mistake
- 1Inequalityx − 4 ≥ 9
- 2Inverse operationAdd 4 to both sides
- 3Simplifyx ≥ 13
- 4GraphOpen circle at 13, shade right
Which step has the mistake? Explain what went wrong, then show the correct work.
Correct workCorrect answer:
Explain your thinkingHow is solving an inequality similar to and different from solving an equation?
Lesson 8.6Solve and Graph Inequalities
ChallengeExtension
Learning target I can solve an inequality and graph its solution set on a number line.
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1Complete the table. Show how you found each value.
Solve each inequality and verify by testing two values.
Inequality Solution Test x = 5 Test x = 20 x + 8 > 15 x − 6 ≤ 10 Show your work
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2Find the mistake. Explain it, then show the correct work.
Find the Solving Error
- 1Inequalityx − 8 ≥ 15
- 2Inverse operationSubtract 8 from both sides
- 3Simplifyx ≥ 7
- 4GraphClosed circle at 7, shade right
Which step has the mistake? Explain what went wrong, then show the correct work.
Correct workCorrect answer: -
3Answer in complete sentences.
A detective has a budget of at most $100 for supplies. She already spent $37 on gloves. Write an inequality for the remaining amount r she can spend, solve it, graph it, and name three possible values for r.
Write your answer