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.
Why This Matters Detective Kim knows that the suspect withdrew more than $50 from an ATM. The amount plus a $3 fee totaled more than $53. She writes x + 3 > 53 and needs to solve for x, then graph the solution set to share with the bank. How can she find and display all possible withdrawal amounts?
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 out!A 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)
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.
Why This Matters Detective Kim knows that the suspect withdrew more than $50 from an ATM. The amount plus a $3 fee totaled more than $53. She writes x + 3 > 53 and needs to solve for x, then graph the solution set to share with the bank. How can she find and display all possible withdrawal amounts?
-
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
-
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
-
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.
Why This Matters Detective Kim knows that the suspect withdrew more than $50 from an ATM. The amount plus a $3 fee totaled more than $53. She writes x + 3 > 53 and needs to solve for x, then graph the solution set to share with the bank. How can she find and display all possible withdrawal amounts?
-
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 − 5 ≥ 3
- x + 1 ≤ 15
- x + 3 > 15
- x − 2 ≥ 9
- x + 6 < 10
-
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
-
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
-
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.
Why This Matters Detective Kim knows that the suspect withdrew more than $50 from an ATM. The amount plus a $3 fee totaled more than $53. She writes x + 3 > 53 and needs to solve for x, then graph the solution set to share with the bank. How can she find and display all possible withdrawal amounts?
-
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
-
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