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
Level 0Most support · every hint printed
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?
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!
-
1Circle the letter of the best answer. Show how you know.
Solve: x + 6 > 14
- Ax > 8
- Bx < 8
- Cx = 8
- Dx > 20
Steps to try- Solve it like an equation: x has 6 added to it. The > symbol just rides along.
- Subtract 6 from both sides — the inequality symbol keeps pointing the same way when you subtract.
- Compute 14 − 6 and keep the >. Then test: pick a number bigger than your result and check it makes x + 6 > 14 true.
Show your work -
2Circle the letter of the best answer. Show how you know.
Solve: x − 9 ≤ 3
- Ax ≤ 6
- Bx ≤ 12
- Cx ≥ 12
- Dx = 12
Steps to try- x has 9 subtracted from it. Undo that first; the ≤ symbol keeps its direction.
- Add 9 to both sides. Adding the same number to both sides never flips the symbol.
- Compute 3 + 9 and keep the ≤. Check by testing a value at or below your boundary in the original inequality.
Show your work -
3Circle the letter of the best answer. Show how you know.
Solve: x + 10 ≥ 25
- Ax ≥ 15
- Bx ≤ 15
- Cx = 15
- Dx ≥ 35
Steps to try- What's being done to x? 10 is added. Plan the inverse operation before you look at the choices.
- Subtract 10 from both sides; the ≥ stays exactly as it is.
- Compute 25 − 10 and keep the ≥. Test your boundary value — it should make both sides of the original exactly equal.
Show your work -
4Circle the letter of the best answer. Show how you know.
Solve: x − 4 < 11
- Ax < 7
- Bx < 15
- Cx > 15
- Dx = 15
Steps to try- x has 4 taken away. Which operation puts that 4 back?
- Add 4 to both sides — the < direction does not change when you add.
- Compute 11 + 4 and keep the <. Check: a number just below your boundary should make x − 4 < 11 true.
Show your work
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 ___.