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6.EE.B.8Common CoreMathExpressions and EquationsGrade 6

6.EE.B.8: Writing and Graphing Inequalities on a Number Line

In plain English: 6.EE.B.8 is the Common Core grade 6 math standard that asks students to write an inequality such as x greater than c, or x less than c, for a limit in a real-world or math problem. Students learn that such an inequality has infinitely many solutions and show all of them on a number line with an open circle and shading. It leads to solving inequalities in grade 7.

Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

Common Core State Standards for Mathematics · Domain: Expressions and Equations (EE) · Cluster: Reason about and solve one-variable equations and inequalities.
Also written as 6.EE.8 · Official standard
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01

Lesson Plan

60 min

Overview

Students write inequalities (statements that one amount is greater than or less than another) of the form x > c and x < c to represent a constraint (a limit or condition) in a real-world or math problem, such as "riders must be taller than 48 inches". The number c is called the boundary number: it separates the solutions from the numbers that are not solutions. Students learn that an inequality like this has infinitely many solutions (so many that the list never ends), not just the whole numbers that come to mind first, and that c itself is not a solution.

Students then show all the solutions at once on a number line diagram: an open circle (an empty circle) at the boundary number and shading with an arrow in the direction of the solutions. Boundary numbers can be whole numbers, fractions, decimals or negative numbers, as in the rest of grade 6. The lesson uses only the symbols > and <, as the standard does. Solving inequalities such as 2x + 3 > 9 waits until grade 7.

Learning Objectives

By the end of this lesson, students will be able to:

  • Write an inequality of the form x > c or x < c for a constraint or condition in a real-world or math problem
  • Test whether a number is a solution of x > c or x < c, including the boundary number itself
  • Explain why an inequality of the form x > c or x < c has infinitely many solutions
  • Represent the solutions of x > c and x < c on a number line with an open circle and shading

Prior Knowledge Required

Students should already be comfortable with:

  • Placing whole numbers, fractions, decimals and negative numbers on a number line 6.NS.C.6
  • Reading statements of order, such as -3 > -7, as positions on a number line 6.NS.C.7
  • Using substitution to test whether a number makes an equation or inequality true 6.EE.B.5
  • Using a letter to stand for an unknown number 6.EE.B.6

Lesson Procedure

60-60 minutes of class time across 5 phases.

  1. Warm-Up10 minutes

    Write this rule on the board: "To win a prize, your paper airplane must fly farther than 4 meters." Ask students to answer on a sticky note.

    Warm-Up Prompt

    "Name five distances that win a prize. Does a flight of 4 meters win? Does 4.5 meters? What is the shortest winning distance? How many winning distances are there?"

    Collect answers and list them on a number line. Students usually start with 5, 6, 7 and 10. Push on 4.5, 4.1 and 4.01: all of them are greater than 4. A flight of exactly 4 meters does not win, because 4 is not greater than 4. Ask for the smallest winning number. Every number a student names, such as 4.001, has a smaller one between it and 4, such as 4.0001, so there is no smallest one, and the list never ends. This is the main idea of the lesson: a statement such as "greater than 4" has infinitely many solutions (so many that you could never finish listing them).

  2. Direct Instruction20 minutes

    Key words. An inequality is a statement that one amount is greater than or less than another. The symbol > means "is greater than", and < means "is less than". The open side of the symbol faces the larger number. A constraint is a limit or condition in a problem, such as "taller than 48 inches". A solution of an inequality is any value of the variable that makes it true. In x > c or x < c, the number c is the boundary number: it separates the solutions from the numbers that are not solutions.

    Graphing on a number line. Draw an open circle (an empty circle) at the boundary number, because c itself is not a solution of x > c or x < c. Then shade the number line in the direction of the solutions and add an arrow: to the right for >, to the left for <. The arrow shows that the solutions go on forever. A filled-in circle means the boundary number is included; that is used for "at least" and "at most" (≥ and ≤), which go beyond this standard. Diagram 1 shows two graphs, and Diagram 2 shows a real rule.

    • Writing x > c for a real-world constraint

      To ride a roller coaster, a rider must be taller than 48 inches. Write an inequality for the heights h that are allowed.

      Equation: h > 48. Solutions include 48.5, 52 and 60 inches; 48 is not a solution (Diagram 2)

    • Writing a less-than inequality with a negative boundary number

      The weather report says tonight's temperature will stay below -3 °C. Write an inequality for the temperature t.

      Equation: t < -3. Solutions include -3.5, -6 and -12; -3 and -2 are not solutions, because they are not less than -3

    • Recognizing infinitely many solutions

      List solutions of x > 3. Can you ever finish the list?

      Equation: 3.1, 3.01, 3.001, 4, 5 1/2 and 1,000 are all solutions. Between 3 and 4 alone there are infinitely many, so the list never ends

    • Graphing a less-than inequality on a number line

      Graph x < 1 1/2.

      Equation: Open circle at 1 1/2, shaded to the left with an arrow. Solutions include 1.4, 0 and -2.5 (Diagram 1)

    • Reading a graph and the reversed form

      A number line has an open circle at -2 and is shaded to the right. Which inequality does it show? What does 7 < y mean?

      Equation: The graph shows x > -2. The statement 7 < y means the same as y > 7: y is greater than 7

    1. Find the boundary number: look for the number in the condition (48 inches, -3 °C).
    2. Choose the symbol: words such as more than, greater than, taller, longer, older, warmer and above give >. Words such as less than, fewer than, under, below, shorter and colder give <.
    3. Test one number: pick a number on the side you think is right and check that it makes the inequality true. Then graph with an open circle and shading.

  3. Guided Practice15 minutes

    Pairs work through four problems, one at a time. For each one they write the inequality, test one solution and one non-solution, and sketch the graph. After each problem, one pair shows its graph.

    Guided practice problems (answers for the teacher)
    ProblemInequalityGraph
    1. A package ships at the lower price if it weighs less than 2 pounds.w < 2Open circle at 2, shaded left
    2. Jada wants to spend less than $25 on a gift.m < 25Open circle at 25, shaded left
    3. Graph y > -4 and name three solutions.y > -4 (for example -3.5, 0 and 10)Open circle at -4, shaded right
    4. A number line has an open circle at 0.5 and is shaded to the left. Write the inequality.x < 0.5Given

    Ask about problem 1: is 2 pounds a solution? (No, 2 is not less than 2.) Is 0.3 pounds? (Yes.) Listen for students who draw the shading toward the wrong side, and ask them to test a number from the shaded part.

  4. Independent Practice10 minutes

    Students solve five problems on their own. (1) A fish tank heater keeps the water warmer than 75 °F. Write the inequality. (t > 75.) (2) Graph n < -1. (Open circle at -1, shaded left.) (3) Name three solutions of x > 2/3 that are less than 1. (For example 3/4, 0.7 and 0.9.) (4) Which of 4, 4.2, 4.19 and 5 are solutions of n < 4.2? (4 and 4.19.) (5) A graph has an open circle at 10 and is shaded to the right. Write the inequality. (x > 10.)

  5. Closure5 minutes

    Exit ticket: (1) Children younger than 12 pay a lower price at a museum. Write an inequality for the age a, and graph it. (a < 12; open circle at 12, shaded left.) (2) Explain why x > 6 has infinitely many solutions. (Any number greater than 6 works, such as 6.5, 6.05 and 100, and you can always name another one.) (3) Is 6 a solution of x > 6? (No, 6 is not greater than 6.)

Differentiation Strategies

For Struggling Students

  • Give a word bank with two columns: words that mean > (more than, over, above, taller) and words that mean < (less than, under, below, shorter)
  • Have students test one number on each side of the boundary number and mark it with a check or a cross before shading
  • Use number lines printed with tick marks every 1 unit, so students only have to place the boundary number

For Advanced Students

  • Ask students to find a number between 0.2 and 0.21 and explain what that shows about the solutions of x > 0.2
  • Give a story whose variable can only be a whole number, such as the number of people on a trip, and ask which solutions of the inequality make sense
  • Ask students to write the same condition two ways, such as t < 5 and 5 > t, and explain why both are correct

Assessment Guidance

What to Look For

Check that students pick the right symbol from the words of the constraint and name the variable with units. They should test at least one number, and they should say that the boundary number is not a solution. On graphs, look for an open circle at the boundary number, shading on the correct side and an arrow. When students explain "infinitely many", listen for non-whole numbers, such as 4.01, not only whole numbers.

02

Classroom Activities

3 Activities

1

Human Number Line

15 minWhole class

Tape a number line from -5 to 5 on the floor. Ten students each hold a number card and stand at their number. The teacher calls out an inequality, and every student whose number is a solution steps forward.

Number Cards (10 cards)

-4.5, -3, -2, -1/2, 0, 3/4, 1, 2.5, 3, 4.9

Procedure

  • Call x > 1. Solutions step forward: 2.5, 3 and 4.9
  • Call x < -1/2. Solutions step forward: -4.5, -3 and -2
  • Call x > -3. Eight cards step forward: every card except -4.5 and -3
  • Call x < 2.5. Seven cards step forward: -4.5 through 1
  • After each call, a student who is not holding a card places an open circle (a paper plate) at the boundary number and lays a strip of paper in the direction of the solutions

Discussion Questions

  • For x > 1, why did the student holding 1 stay back?
  • For x > -3, the student holding -3 stayed back, but -2 stepped forward. Why?
  • How many more cards could we add that would step forward for x > 1? (Infinitely many, such as 1.5, 1.05 and 20)

Modification for Small Spaces

Use a number line drawn on the board and sticky notes instead of students. Each pair places the sticky notes that are solutions above the line.

2

Constraint Card Sort

20 minGroups of 3-4

Each group gets 8 situation cards, 8 inequality cards and 8 number line cards. Groups match each situation to its inequality and graph, then test one solution for each match.

Situation Cards (8 cards)

  • A. A large envelope needs a package rate if it weighs more than 13 ounces. (w > 13)
  • B. The outdoor pool opens when the air is warmer than 20 °C. (t > 20)
  • C. A tomato plant grows best with more than 6 hours of sun a day. (h > 6)
  • D. A freezer must stay colder than -15 °C. (t < -15)
  • E. A summer camp group is for children older than 8. (a > 8)
  • F. A phone shows a warning when its battery is below 20%. (b < 20)
  • G. A pencil fits in the case if it is shorter than 19 cm. (p < 19)
  • H. A thermometer at a research station in Antarctica read below -40 °C. (t < -40)

Procedure

  • Match each situation card with one inequality card and one number line card
  • Sort the matches into two columns: x > c and x < c
  • For each match, write one number that is a solution and one that is not on a sticky note
  • Trade sticky notes with another group and check each other's numbers

Discussion Questions

  • Which cards have a negative boundary number? (D and H)
  • Which cards use >? (A, B, C and E) Which use
  • On card G, is a pencil that is exactly 19 cm long a solution? Why or why not?
  • On card E, ages are often counted in whole years. Does that change how many solutions the inequality a > 8 has?

Challenge Variation

Groups write two new situation cards from their own school day, one for each symbol, and trade them with another group to match and graph.

3

Measure and Fit

15 minPairs

Pairs measure pencils and decide which ones fit a pencil case that holds only pencils shorter than 19 cm. They graph the inequality on grid paper and mark each pencil on it.

Procedure

  • Collect 5 pencils of different lengths. Measure each to the nearest 0.1 cm with a centimeter ruler. Used pencils are usually 8-18 cm long, and a new pencil is about 19 cm
  • Draw a number line from 5 to 20 cm on grid paper. Graph p < 19 with an open circle at 19 and shading to the left
  • Mark each pencil's length with a dot and label it "fits" or "does not fit"
  • Write one more inequality for your desk: "A book fits on the shelf under the desk if it is shorter than ___ cm". Measure the shelf height to fill in the blank, then graph it

Discussion Questions

  • Two pencils measure 12.3 cm and 12.35 cm. Are both solutions of p < 19? Could there be a length between them?
  • Why does the shading go all the way to the arrow, even though no pencil is shorter than 1 cm?
  • A new pencil measures 19.0 cm. Does it fit? What does the open circle tell you?

03

Diagrams & Visual Aids

2 diagrams

Diagram 1: Graphing Greater-Than and Less-Than Inequalities

x > 3: an open circle at 3, shaded to the right (every number greater than 3) -3 -2 -1 0 1 2 3 4 5 6 3.1 4 5 1/2 boundary number 3 (not a solution) x < 1 1/2: an open circle at 1 1/2, shaded to the left (every number less than 1 1/2) -3 -2 -1 0 1 2 3 4 5 6 -2.5 0 1.4 boundary number 1 1/2 (not a solution) Dots show a few of the infinitely many solutions.
Top: the graph of x > 3 has an open circle at 3 and is shaded to the right, because every number greater than 3 is a solution. Bottom: the graph of x < 1 1/2 has an open circle at 1 1/2 and is shaded to the left. The dots mark a few solutions, and the arrows show that the solutions go on forever.

Diagram 2: A Real-World Constraint on a Number Line

Ride rule: riders must be taller than 48 inches, so h > 48 40 42 44 46 48 50 52 54 56 Ava: 46 1/2 in, cannot ride Ben: 48 in, cannot ride Cruz: 49 1/2 in, can ride Dee: 53 in, can ride Height (inches) Ben is exactly 48 inches, which is not taller than 48, so he is not a solution. Green: a solution of h > 48 (can ride). Red: not a solution (cannot ride).
A roller coaster allows riders taller than 48 inches, so the allowed heights are h > 48. Cruz (49 1/2 inches) and Dee (53 inches) are solutions. Ava (46 1/2 inches) is not, and neither is Ben, who is exactly 48 inches tall. The heights are invented but typical for children aged 6-9.

04

Homework Assignment

~30 min

6.EE.B.8 Homework: Writing and Graphing Inequalities

Directions: Name the variable and its units for every word problem. Use the symbols > and <. Draw every graph with an open circle at the boundary number, shading and an arrow.

Part 1: Writing Inequalities (Problems 1-2)

  1. Write an inequality for each condition: (a) A skateboard ramp is for riders who weigh less than 90 kg. (b) You earn a reading badge for reading more than 12.5 hours this month. (c) Last night the temperature was below -8 °F.
  2. At a field day, a prize goes to every ball throw longer than 15.5 meters. Write an inequality for the length d of a prize-winning throw. Name three solutions, including one that is not a whole number, and one length that does not win.

Part 2: Infinitely Many Solutions (Problems 3-4)

  1. For x < 5/6: (a) name four solutions: one negative number, one fraction, one decimal and one whole number. (b) Is 5/6 a solution? Explain. (c) Explain why you could keep naming solutions forever.
  2. Which of these numbers are solutions of y > -2.5? -3, -2.5, -2.4, 0, 1/4, -10. Then explain whether your list shows all the solutions.

Part 3: Number Line Diagrams (Problems 5-6)

  1. Graph each inequality on its own number line: (a) x > 1.75 (b) x < -4 (c) x > 0. Label the boundary number on each graph.
  2. A school needs a second van for a field trip if more than 7 people go. Write an inequality for the number of people p, and graph it. Then explain which solutions make sense in this story.

Rubric

CriterionFull Credit (2 pts)Partial Credit (1 pt)No Credit (0 pts)
Writing InequalitiesCorrect symbol and boundary number, and the variable is named with unitsCorrect inequality, but the variable is not namedWrong symbol or boundary number
Testing SolutionsAll solutions correct, and the boundary number is correctly excludedOne error, or the boundary number is counted as a solutionTwo or more errors
Infinitely Many SolutionsClear explanation with non-whole examplesExplanation uses only whole numbersNo explanation
Number Line DiagramsOpen circle at the boundary number, correct shading and an arrow on every graphOne graph with a closed circle or a missing arrowShading on the wrong side

05

Quiz: 20 Questions

Interactive, with answers

Instructions

Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.

Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.

0 of 20 answered · 0 correct

  1. Question 1 of 20 · Multiple Choice

    The temperature stayed colder than 10 °F all day. Which inequality describes the temperature t?

  2. Question 2 of 20 · Multiple Choice

    A phone plan charges extra if you use more than 5 GB of data in a month. Which inequality describes the data d, in GB, that costs extra?

  3. Question 3 of 20 · Multiple Choice

    Which number line shows the solutions of x < 3.5?

  4. Question 4 of 20 · Multiple Choice

    Which number is a solution of x > -7?

  5. Question 5 of 20 · Multiple Choice

    How many solutions does the inequality x < 7 have?

  6. Question 6 of 20 · Multiple Choice

    Which number is NOT a solution of y < 0.3?

  7. Question 7 of 20 · Multiple Choice

    A number line has an open circle at -6 and is shaded to the right. Which inequality does it show?

  8. Question 8 of 20 · Multiple Choice

    Which inequality means the same as 5 < x?

  9. Question 9 of 20 · Multiple Choice

    Which statement about the inequality x > 2.5 is true?

  10. Question 10 of 20 · Multiple Choice

    A tourist submarine dives deeper than 25 meters below sea level. Elevations below sea level are negative. Which inequality describes the submarine's elevation e, in meters?

  11. Question 11 of 20 · Multiple Choice

    Which situation can be represented by x > 1.5?

  12. Question 12 of 20 · Multiple Choice

    Which set contains only solutions of x < -2?

  13. Question 13 of 20 · Multiple Choice

    A board game needs more than 4 players. The inequality p > 4 describes the number of players p. Which list shows the solutions that make sense in this story?

  14. Question 14 of 20 · Multiple Choice

    A number line has an open circle at 3/4 and is shaded to the left. Which inequality does it show?

  15. Question 15 of 20 · Short Answer

    A bag of flour passes the factory check if it weighs more than 2.2 kg. Write an inequality for the weight w of a bag that passes, and name two solutions.

  16. Question 16 of 20 · Short Answer

    Describe the graph of x < -1.5 on a number line, and name three solutions.

  17. Question 17 of 20 · Short Answer

    Explain why the inequality x > 9 has infinitely many solutions. Give at least three solutions, including one that is not a whole number.

  18. Question 18 of 20 · Short Answer

    Is 12 a solution of n > 12? Explain.

  19. Question 19 of 20 · Short Answer

    In a desert town, the afternoon temperature was higher than 40 °C. Write an inequality for the temperature t, and describe its graph.

  20. Question 20 of 20 · Short Answer

    A student graphs x < 4 with an open circle at 4 and shading to the right. Explain the error and describe the correct graph.

0 of 20 answered · 0 correct

06

Frequently Asked Questions

10 Questions

What does 6.EE.B.8 mean?

6.EE.B.8 means students write inequalities such as x > 5 or x < 5 to describe a limit in a problem, understand that these inequalities have infinitely many solutions, and show those solutions on a number line. For example, "a carry-on bag must weigh less than 10 kg" becomes w < 10.

Is 6.EE.B.8 about solving inequalities?

No, not in the grade 7 sense. In 6.EE.B.8, students write and graph inequalities that already have x alone on one side. Solving inequalities with steps, such as 3x + 2 > 11, belongs to grade 7 (7.EE.B.4).

Why does an inequality like x greater than 2 have infinitely many solutions?

Because there is no end to the numbers greater than 2. For x > 2, the numbers 3, 10 and 1,000 work, and so do 2.1, 2.01 and 2.001. Between any two numbers there is always another one, so the list can never be finished.

Why do we use an open circle on the number line?

An open circle shows that the boundary number is not a solution. In x < 8, the number 8 is not less than 8, so it is not included. A filled-in circle is used for ≥ and ≤ ("at least" and "at most"), which students meet in the same year in many classes but which go beyond this standard's x > c and x < c.

How do students know which way to shade?

They test a number. Pick a number on one side of the boundary and substitute it. If the inequality is true, shade that side. For x > c, the solutions are always to the right of c, and for x < c they are always to the left.

What are common mistakes with inequalities in grade 6?

A common mistake is including the boundary number as a solution. Another is shading in the wrong direction, especially with negative numbers: many students think -6 is greater than -2. A third is reading a statement such as 3 < x backward. Having students test one number catches all three.

What words in a problem signal greater than or less than?

More than, greater than, over, above, taller, longer, older and warmer usually signal >. Less than, fewer than, under, below, shorter, younger and colder usually signal <. Students should still check the meaning: "more than 10 m below sea level" means an elevation less than -10.

What if only whole numbers make sense in the story?

The inequality still has infinitely many solutions, but the story may allow only some of them. For "fewer than 30 students on a bus", s < 30 includes 29.5, yet only the whole numbers up to 29 make sense for students. Asking which solutions fit the story is a good discussion question.

Can the boundary number be negative or a fraction?

Yes. The standard does not limit c. Boundary numbers like -3, 1 1/2 and 0.75 are all fair, and they give students practice with the number line skills of 6.NS.C.6 and 6.NS.C.7.

How does 6.EE.B.8 connect to high school math?

Writing a constraint as an inequality is the first step of a skill used all through algebra. In grade 7, students solve inequalities and graph their solutions. In high school, students write inequalities for constraints in problems and decide which solutions are reasonable (HSA.CED.A.3).