Lesson 8.5Testing Solutions
Start hereWords, worked example, and sentence starters
Learning target I can graph the solutions of an inequality on a number line.
Start hereWords for this lesson
Read each word before you begin. Say it out loud.
| Word | What it means | Example |
|---|---|---|
| Number lineSpanish: Recta numérica | A straight line where numbers are placed in order; numbers get smaller to the left and larger to the right. | A horizontal line with marks at 0, 1, 2, 3, ... showing where values fall |
| Open circleSpanish: Círculo abierto | An empty circle showing a number is NOT included. | used for < and > |
| Closed circleSpanish: Círculo cerrado | A filled-in circle showing a number IS included. | x ≥ 5: closed circle at 5 means 5 IS a solution, and so are 6, 7, 8, ... |
| Solution setSpanish: Conjunto solución | All the numbers that make the inequality true. | For x > 3, the solution set includes 3.1, 4, 5, 10, 100 — any number greater than 3 |
| InequalitySpanish: Desigualdad | A math sentence that compares two sides with <, >, ≤, or ≥. | x ≥ 3 is an inequality; it is graphed as a closed circle on 3 with an arrow to the right. |
| Boundary pointSpanish: Punto límite | The number on the number line where the solution set starts. | the 5 in x > 5 |
How it worksWorked exampletesting values, then graphing with an open dot
These numbers are not on your problems. The steps are. Follow them with your own numbers.
A rocket club launches only when the wind speed is less than 14 miles per hour. Test and graph.
- Write the inequality: let s = the wind speed, so s < 14.
- Substitute 11: the statement 11 < 14 is true, so 11 is a solution.
- Substitute the boundary 14: the statement 14 < 14 is false, so it is not a solution.
- Graph it: an OPEN dot at 14 and an arrow pointing left toward slower speeds.
Answer: s < 14: open dot at 14 with an arrow left; 11 is a solution and 14 is not.
Now you trySame steps, your turn
The steps are the same as the worked example. The numbers are yours. Do the work.
A pool opens the deep end when the water is warmer than 68 degrees. Test values, then graph.
- Write the inequality: let t = , so t 68.
- Substitute 72. Is the statement true? Is 72 a solution?
- Substitute the boundary 68. True or false? A solution?
- Graph it: dot at 68, arrow pointing . Draw the number line.
Say it and write itSentence starters
Finish each sentence out loud with a partner. Then use them in your writing.
- I substituted and the statement was .
- The value (is / is not) a solution because .
- I used a dot at because .
- My arrow points because the solutions are .
Word bank inequalitysubstitutetruefalsesolutionopen circleclosed circleboundary pointarrownumber linetestgreater thanless than
Watch outA common mistake
A common mistake in Graph Inequalities is treating ≥ and ≤ the same as > and < and drawing the wrong circle. For example, students often graph x ≥ 6 with an OPEN circle (as if it were x > 6), which wrongly excludes 6 even though 6 itself makes x ≥ 6 true. A second common mistake is choosing the right circle but shading the wrong direction — for x < 9 students sometimes shade right instead of left. Always check the symbol twice: once for the circle (does it have an 'or equal to' line?) and once for the shading direction (does the symbol point toward larger or smaller numbers?).
Lesson 8.5Testing Solutions
Apply Day · Version ASupported practice
Learning target I can graph the solutions of an inequality on a number line.
-
1Complete each step, then write the final answer.
Graph x ≥ 14. Which circle do you draw at 14: open or closed?
Show your workHint Does ≥ include 14? Is 14 ≥ 14 true?
Final answer: -
2Complete each step, then write the final answer.
Graph x < 11. Which way do you shade from 11: left or right?
Show your workHint Numbers less than 11 are on which side of 11?
Final answer: -
3Complete each step, then write the final answer.
Graph x > 1. Which way do you shade from 1: left or right?
Show your workHint Numbers greater than 1 are on which side of 1?
Final answer:
Lesson 8.5Testing Solutions
Apply Day · Version BCore practice
Learning target I can graph the solutions of an inequality on a number line.
-
1Circle the letter of the best answer. Show how you know.
Open circle at 4, shaded to the left. The number line shows an open circle at 4, shaded to the left. Which inequality does it show?
- Ax < 4
- Bx ≤ 4
- Cx > 4
- Dx ≥ 4
Show your work -
2Complete each step, then write the final answer.
Graph 1.5 < x. Which way do you shade from 1.5: left or right?
Show your workFinal answer: -
3Complete each step, then write the final answer.
Graph x < 9. Which way do you shade from 9: left or right?
Show your workFinal answer:
Lesson 8.5Testing Solutions
Apply Day · ChallengeExtension
Learning target I can graph the solutions of an inequality on a number line.
-
1Circle the letter of the best answer. Show how you know.
A number line shows an open circle at 7, and the shading covers the numbers to the right of 7. Which inequality does this graph represent?
- Ax > 7
- Bx ≥ 7
- Cx < 7
- Dx ≤ 7
Show your work -
2Complete each step, then write the final answer.
Mo graphs 5 > x with an open circle at 5. He shades right because > points right. Which way should he shade: left or right?
Show your workFinal answer: -
3Complete each step, then write the final answer.
Graph 20 ≥ k. Which way do you shade from 20: left or right?
Show your workFinal answer: