Corequisite Skills for Introductory StatisticsXYZ Homework Edition

2.3 Inequalities as Intervals on the Number Line

Narrated lesson: Inequalities as Intervals on the Number Line

Synthetic narration with English captions, a transcript, and time to think before checking the answer.

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Graph Inequalities on the Number Line

Do you remember what it means for a number to be a solution to an equation? A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

What about the solution of an inequality? What number would make the inequality x>3 true? Are you thinking, ‘x could be 4’? That’s correct, but x could be 5 too, or 20, or even 3.001. Any number greater than 3 is a solution to the inequality x>3.

We show the solutions to the inequality x>3 on the number line by shading in all the numbers to the right of 3, to show that all numbers greater than 3 are solutions. Because the number 3 itself is not a solution, we put an open parenthesis at 3. The graph of x>3 is shown in Figure 2.7. In these graphs, the thick coloured ray marks the solution set, and its arrow shows the direction in which the solutions continue.

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a dark-teal line extending to the right of the parenthesis.
This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a dark-teal line extending to the right of the parenthesis.
Figure 2.7 The inequality x>3 is graphed on this number line.

The graph of the inequality x≥3 is very much like the graph of x>3, but now we need to show that 3 is a solution, too. We do that by putting a bracket at x=3, as shown in Figure 2.8.

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 3 is graphed on the number line, with an open bracket at x equals 3, and a dark-teal line extending to the right of the bracket.
This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 3 is graphed on the number line, with an open bracket at x equals 3, and a dark-teal line extending to the right of the bracket.
Figure 2.8 The inequality x≥3 is graphed on this number line.

Notice that the open parentheses symbol, (, shows that the endpoint of the inequality is not included. The open bracket symbol, [, shows that the endpoint is included.

We can also represent inequalities using interval notation. As we saw above, the inequality x>3 means all numbers greater than 3. There is no upper end to the solution to this inequality. In interval notation, we express x>3 as (3,∞). The symbol ∞ is read as ‘infinity’. It is not an actual number. Figure 2.9 shows both the number line and the interval notation.

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a dark-teal line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, 3 comma infinity, parenthesis.
This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a dark-teal line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, 3 comma infinity, parenthesis.
Figure 2.9 The inequality x>3 is graphed on this number line and written in interval notation.

The inequality x≤1 means all numbers less than or equal to 1. There is no lower end to those numbers. We write x≤1 in interval notation as (−∞,1]. The symbol −∞ is read as ‘negative infinity’. Figure 2.10 shows both the number line and interval notation.

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to 1 is graphed on the number line, with an open bracket at x equals 1, and a dark-teal line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 1, bracket.
This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to 1 is graphed on the number line, with an open bracket at x equals 1, and a dark-teal line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 1, bracket.
Figure 2.10 The inequality x≤1 is graphed on this number line and written in interval notation.

Did you notice how the parenthesis or bracket in the interval notation matches the symbol at the endpoint of the arrow? These relationships are shown in Figure 2.11.

This figure shows the same four number lines as above, with the same interval notation labels. Below the interval notation for each number line, there is text indicating how the notation on the number lines is similar to the interval notation. The first number line is a graph of x is greater than a, and the interval notation is parenthesis, a comma infinity, parenthesis. The text below reads: “Both have a left parenthesis.” The second number line is a graph of x is greater than or equal to a, and the interval notation is bracket, a comma infinity, parenthesis. The text below reads: “Both have a left bracket.” The third number line is a graph of x is less than a, and the interval notation is parenthesis, negative infinity comma a, parenthesis. The text below reads: “Both have a right parenthesis.” The last number line is a graph of x is less than or equal to a, and the interval notation is parenthesis, negative infinity comma a, bracket. The text below reads: “Both have a right bracket.”
This figure shows the same four number lines as above, with the same interval notation labels. Below the interval notation for each number line, there is text indicating how the notation on the number lines is similar to the interval notation. The first number line is a graph of x is greater than a, and the interval notation is parenthesis, a comma infinity, parenthesis. The text below reads: “Both have a left parenthesis.” The second number line is a graph of x is greater than or equal to a, and the interval notation is bracket, a comma infinity, parenthesis. The text below reads: “Both have a left bracket.” The third number line is a graph of x is less than a, and the interval notation is parenthesis, negative infinity comma a, parenthesis. The text below reads: “Both have a right parenthesis.” The last number line is a graph of x is less than or equal to a, and the interval notation is parenthesis, negative infinity comma a, bracket. The text below reads: “Both have a right bracket.”
Figure 2.11 The notation for inequalities on a number line and in interval notation use similar symbols to express the endpoints of intervals.

Section Exercises

Practice Makes Perfect

Graph Inequalities on the Number Line

In the following exercises, graph each inequality on the number line.

ⓐ x≤−2
ⓑ x>−1ⓒ x<0

ⓐ x>1ⓑ x<−2
ⓒ x≥−3

Show solution
  1. ⓐ
    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 1 is graphed on the number line, with an open parenthesis at x equals 1, and a dark line extending to the right of the parenthesis.
  2. ⓑ
    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 2 is graphed on the number line, with an open parenthesis at x equals negative 2, and a dark line extending to the left of the parenthesis.
  3. ⓒ
    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to negative 3 is graphed on the number line, with an open bracket at x equals negative 3, and a dark line extending to the right of the bracket.

ⓐ x≥−3ⓑ x<4ⓒ x≤−2

ⓐ x≤0ⓑ x>−4ⓒ x≥−1

Show solution
  1. ⓐ
    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to 0 is graphed on the number line, with an open bracket at x equals 0, and a dark line extending to the left of the bracket.
  2. ⓑ
    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than negative 4 is graphed on the number line, with an open parenthesis at x equals negative 4, and a dark line extending to the right of the parenthesis.
  3. ⓒ
    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to negative 1 is graphed on the number line, with an open bracket at x equals negative 1, and a dark line extending to the right of the bracket.

In the following exercises, graph each inequality on the number line and write in interval notation.

ⓐ x<−2ⓑ x≥−3.5ⓒ x≤23

ⓐ x>3ⓑ x≤−0.5ⓒ x≥13

Show solution
  1. ⓐ
    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a dark line extending to the right of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, 3 comma infinity, parenthesis.
  2. ⓑ
    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to negative 0.5 is graphed on the number line, with an open bracket at x equals negative 0.5, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 0.5, bracket.
  3. ⓒ
    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 1/3 is graphed on the number line, with an open bracket at x equals 1/3 (written in), and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, 1/3 comma infinity, parenthesis.

ⓐ x≥−4ⓑ x<2.5ⓒ x>−32

ⓐ x≤5ⓑ x≥−1.5ⓒ x<−73

Show solution
  1. ⓐ
    This figure is a number line with tick marks. The inequality x is less than or equal to 5 is graphed on the number line, with an open bracket at x equals 5, and a dark line extending to the left of the bracket. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 5, bracket.
  2. ⓑ
    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to negative 1.5 is graphed on the number line, with an open bracket at x equals negative 1.5, and a dark line extending to the right of the bracket. Below the number line is the solution written in interval notation: bracket, negative 1.5 comma infinity, parenthesis.
  3. ⓒ
    This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than negative 7/3 is graphed on the number line, with an open parenthesis at x equals negative 7/3 (written in), and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma negative 7/3, parenthesis.

Corequisite Skills for Introductory Statistics is a custom course edition compiled for Introductory Statistics corequisite courses from OpenStax Contemporary Mathematics, Elementary Algebra 2e, Intermediate Algebra 2e, Prealgebra 2e. Content by OpenStax is reproduced as published, apart from any corrections recorded for this edition. Sections are renumbered to follow the course outline. The course outline is adapted from Larry Green's Support Course for Elementary Statistics (LibreTexts, https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Support_Course_for_Elementary_Statistics), used under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Lessons 2.4 and 6.7 are written for this edition by XYZ Homework. Every OpenStax lesson is abridged to the subsections and practice exercises this course uses. Lessons are headed with their course titles, not the source section titles. Access the complete originals free at openstax.org. License: CC-BY-NC-SA-4.0.

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