Neft Teacher · Learn It

Unit 8 · Standard 6.AT.9

Understand Inequalities and Their Solutions

Lesson 8-5 · Learn It — how the math works

Why we're learning this

Detective Okafor is building a timeline. The suspect was seen at the scene no earlier than 2:00 PM, meaning the arrival time t satisfies t ≥ 2. She needs to plot this on a timeline to share with her team. How should she mark the number line?

✱ Math Workbench

Learn It

How do I graph an inequality on a number line?

Graphing an inequality shows every value that makes it true. I draw a circle at the boundary number and shade the side that works. The circle tells me if the boundary is included.

Picture itSee what the math looks like before you read about it

Four number lines are shown, and every one of them is marked at 5. Two use an open circle and two use a filled-in circle, and the shading runs a different direction on each line.
Where you see this in real lifeFour number lines are shown, and every one of them is marked at 5. Two use an open circle and two use a filled-in circle, and the shading runs a different direction on each line.

Watch me solve itI do — read one step at a time

The problem looks like this
A number line graphing t ≥ 2: a closed circle at 2 with the line shaded to the right. -1012345 t ≥ 2 A filled circle means 2 is included. The shading goesright.
  1. Graph t ≥ 2. The boundary number is 2.
  2. The symbol ≥ includes 2, so I draw a closed (filled) circle at 2.
  3. 'Greater than or equal to' means larger values work, so I shade to the right.
  4. Now the graph shows 2, 3, 4, and every value above 2.

Watch out — mistakes students make here

  • '≥' includes 9, so the circle should be filled/closed.
  • x ≥ 9 means 9 and up — shade toward larger numbers.
  • Both parts are off — '≥' is a closed circle, shaded right.

Try it with meWe do — you answer first, then check

Same steps, new numbers.

  1. Graph x < 5. What number is the boundary?
  2. The symbol < does NOT include 5, so we use an open (empty) circle at 5.
  3. Which way do we shade for 'less than'?
    Check this step
    To the left, toward smaller numbers.

On your ownYou do — solve this one by yourself

Before you start, remember:

  • Graph x > 4, x ≤ 7, and x ≥ 2 by choosing the right circle and shading.
  • Then complete the practice graphs on your own.

Solve: A shows a at 8 and is shaded to the left. Which does this represent?

My answer

Stuck? Open one hint at a time.

Hint 1

Read the graph's two clues: the circle at 8 is CLOSED, and the shading runs LEFT toward smaller numbers.

Hint 2

A closed circle means 8 itself is a solution, so the symbol needs the 'or equal' bar. Shading left means solutions are 8 or smaller.

Hint 3

Test x = 5 (in the shaded part) and x = 8 (the circle): both must make the inequality true. Which choice passes both tests?

Check my answer
✅ Answer: x ≤ 8
Closed circle means the value is included (≤ or ≥). Shaded left means less than or equal to: x ≤ 8.

Turn & TalkSay it out loud with a partner

Detective Okafor knows the suspect arrived 'no earlier than 2:00 PM,' so t ≥ 2. On the timeline, should the mark at 2 be an open or , and which way should she shade?

Try starting with:

  • I use a ___ circle at 2 because 2:00 ___ included.Uso un círculo ___ en 2 porque las 2:00 ___ incluidas.
  • I shade ___ because the times can be ___.Sombreo ___ porque las horas pueden ser ___.
Use these wordsnumber lineopen circleclosed circlesolution set