Inequalities on a Number Line

How to show the solutions of an inequality on a number line with an open or a filled circle and an arrow, and how to read an inequality from a number line.

Before you start

An inequality such as $x > 3$ has infinitely many solutions: 3.1, 4, 10, 1000 and every other number greater than 3. They cannot all be listed, but they can all be shown at once on a number line.

The number 3 itself is not a solution of $x > 3$ , but it is a solution of $x \ge 3$ . The number that separates the solutions from the numbers that are not solutions, 3 here, is the boundary value.

$3 < x$ says the same as $x > 3$ : swap the two sides and turn the sign round with them.

Drawing an Inequality

To show an inequality on a number line, we mark its boundary value with a circle and draw a thick line with an arrow over all its solutions. The circle says whether the boundary value is a solution. The arrow says that the solutions go on without end.

Showing an inequality on a number line (with the unknown on the left)

The circle. Draw an open circle at the boundary value for $<$ or $>$ : the boundary value is not a solution. Draw a filled circle for $\le$ or $\ge$ : the boundary value is a solution.

The arrow. For $>$ or $\ge$ , draw the line and arrow to the right of the circle, over the numbers greater than the boundary value. For $<$ or $\le$ , draw it to the left, over the numbers less than the boundary value.

This rule needs the unknown on the left. If it is on the right, as in $4 < x$ , rewrite the inequality first.

Inequality	Is 3 a solution?	Circle at 3	Line and arrow
$x > 3$	No	open	to the right
$x \ge 3$	Yes	filled	to the right
$x < 3$	No	open	to the left
$x \le 3$	Yes	filled	to the left

Example 1: Drawing four inequalities

Show each inequality on a number line.
(a) $x < 3$
(b) $x \ge 3$
(c) $x > -2$
(d) $x \le 0$

Solution

(a) The sign is $<$ , so 3 is not a solution: an open circle at 3. The solutions are the numbers less than 3, so the line and arrow go to the left.

(b) The sign is $\ge$ , so 3 is a solution: a filled circle at 3. The solutions are 3 and the numbers greater than 3, so the line and arrow go to the right.

(c) An open circle at $-2$ , and the line and arrow to the right. The numbers greater than $-2$ include $-1$ , 0 and 1: to the right of $-2$ , not further from zero.

(d) A filled circle at 0, and the line and arrow to the left, over 0 and every negative number.

(a) open circle at 3, arrow to the left (b) filled circle at 3, arrow to the right (c) open circle at $-2$ , arrow to the right (d) filled circle at 0, arrow to the left. (a) and (b) have the same boundary value, 3, but differ in both the circle and the direction.

Boundary Values That Are Not Whole Numbers

When the boundary value is not a whole number, its circle goes between two marks of the number line, in its right place, with its value written above it.

Example 2: A boundary value between the marks

Show each inequality on a number line.
(a) $x \ge 2.5$
(b) $x < -\frac{3}{2}$

Solution

(a) 2.5 is halfway between 2 and 3. The sign is $\ge$ : a filled circle at 2.5, and the line and arrow to the right.

(b) $-\frac{3}{2} = -1.5$ , which is halfway between $-2$ and $-1$ . The sign is $<$ : an open circle at $-1.5$ , and the line and arrow to the left.

(a) filled circle at 2.5, arrow to the right (b) open circle at $-1.5$ , arrow to the left. In (b) the boundary value may be written as $-1.5$ or $-\frac{3}{2}$ .

When the Unknown Is on the Right

$4 < x$ is read "4 is less than $x$ ": $x$ is greater than 4, so its solutions are to the right of 4, even though the sign is $<$ . Drawing straight from the sign would send the arrow the wrong way. So first rewrite the inequality with the unknown on the left, then draw it.

Example 3: Rewriting before drawing

Show each inequality on a number line.
(a) $4 < x$
(b) $-1 \ge x$

Solution

(a) Swap the sides and turn the sign round: $4 < x$ is $x > 4$ . An open circle at 4, and the line and arrow to the right.

(b) $-1 \ge x$ is $x \le -1$ . A filled circle at $-1$ , and the line and arrow to the left.

(a) $x > 4$ (b) $x \le -1$ , drawn above.

Reading an Inequality from a Number Line

Reading a number line

Step 1: The boundary value is the number at the circle.

Step 2: An arrow to the right shows the numbers greater than it ( $>$ or $\ge$ ). An arrow to the left shows the numbers less than it ( $<$ or $\le$ ).

Step 3: An open circle gives $>$ or $<$ . A filled circle gives $\ge$ or $\le$ .

Example 4: Writing the inequality shown

Write down the inequality in $x$ shown by each number line.

(a)

(b)

(c)

Solution

(a) The boundary value is 1. The arrow goes to the left, so $x$ is less than 1. The circle is open, so 1 is not included.

(b) The boundary value is $-3$ . The arrow goes to the right, so $x$ is greater than $-3$ . The circle is filled, so $-3$ is included.

(a) $x < 1$ (b) $x \ge -3$ (c) $x \le 0.5$

Example 5: Solutions from a number line

The number line shows the solutions of an inequality in $x$ .
(a) Write down two numbers that are solutions and one number that is not.
(b) Is $-2$ a solution? Is $-2.5$ a solution?

Solution

The number line shows $x \ge -2$ .

(a) Any number that the thick line covers is a solution, such as 0 or 2. Any number to the left of $-2$ is not, such as $-3$ .

(b) The circle at $-2$ is filled, so $-2$ is a solution. $-2.5$ is to the left of $-2$ , so it is less than $-2$ and is not a solution.

(a) For example, 0 and 2 are solutions, and $-3$ is not. (b) $-2$ is a solution; $-2.5$ is not.

Spotting Errors and Matching

A drawing can go wrong in three ways: the wrong circle, the wrong direction, or the circle at the wrong number. Check all three each time.

Example 6: Spot the error

Each number line was drawn for the inequality above it, and each is wrong. Say what is wrong, and draw it correctly.

(a) $x > 2$

(b) $x \le -1$

Solution

(a) The circle is filled, but the sign is $>$ , so 2 is not a solution. The circle must be open.

(b) The arrow goes to the right, but $x \le -1$ means $x$ is less than or equal to $-1$ . The line and arrow must go to the left.

(c) The drawing was made from the sign $<$ without rewriting. $4 < x$ is $x > 4$ , so the line and arrow must go to the right.

(a) The circle should be open. (b) The arrow should go to the left. (c) The arrow should go to the right, because $4 < x$ is $x > 4$ .

Example 7: Matching inequalities with number lines

Match each inequality with its number line.
(i) $x \le 3$ (ii) $3 < x$ (iii) $x < 3$ (iv) $x \ge 3$

Solution

All four have the boundary value 3, so look at the arrow and the circle.

(i) $x \le 3$ : arrow to the left, filled circle. That is D.

(ii) $3 < x$ is $x > 3$ : arrow to the right, open circle. That is C.

(iii) $x < 3$ : arrow to the left, open circle. That is B. It differs from D only in the circle.

(iv) $x \ge 3$ : arrow to the right, filled circle. That is A.

(i) D (ii) C (iii) B (iv) A

Common Mistakes

A filled circle for $<$ or $>$ . With $<$ and $>$ the boundary value is not a solution, so its circle is open.
Drawing $4 < x$ to the left. $4 < x$ is $x > 4$ , so its solutions are to the right of 4. Write the unknown on the left before you choose the direction.
Putting the circle for 2.5 on the mark for 2 or 3. It goes halfway between them, with 2.5 written above it.
Reading the arrow for $-2$ the wrong way. The numbers greater than $-2$ are to its right ( $-1$ , 0, 1, …), not further from zero.
Drawing $x < 3$ and $x \le 3$ alike. They have the same arrow, but different circles.

Practice

Draw each number line on paper before you open its solution.

Question 1

Show each inequality on a number line.
(a) $x > 5$
(b) $x \le 1$
(c) $x < -4$
(d) $x \ge 0$

ShowHide solution

(a) An open circle at 5, arrow to the right.

(b) A filled circle at 1, arrow to the left.

(d) A filled circle at 0, arrow to the right.

(a) open circle at 5, arrow to the right (b) filled circle at 1, arrow to the left (c) open circle at $-4$ , arrow to the left (d) filled circle at 0, arrow to the right

Question 2

Show each inequality on a number line.
(a) $x < 0.5$
(b) $x \ge -2.5$
(c) $x > \frac{7}{2}$

ShowHide solution

(a) 0.5 is halfway between 0 and 1. An open circle at 0.5, arrow to the left.

(b) $-2.5$ is halfway between $-3$ and $-2$ . A filled circle at $-2.5$ , arrow to the right.

(a) open circle at 0.5, arrow to the left (b) filled circle at $-2.5$ , arrow to the right (c) open circle at 3.5, arrow to the right

Question 3

Write each inequality with the unknown on the left. Then show it on a number line.
(a) $6 > x$
(b) $-3 \le x$
(c) $0 < x$
(d) $2.5 \ge x$

ShowHide solution

(a) $6 > x$ is $x < 6$ : an open circle at 6, arrow to the left.

(b) $-3 \le x$ is $x \ge -3$ : a filled circle at $-3$ , arrow to the right.

(d) $2.5 \ge x$ is $x \le 2.5$ : a filled circle at 2.5, arrow to the left.

(a) $x < 6$ (b) $x \ge -3$ (c) $x > 0$ (d) $x \le 2.5$ , drawn above.

Question 4

Write down the inequality in $x$ shown by each number line.

(a)

(b)

(c)

(d)

ShowHide solution

(a) Open circle at $-1$ , arrow to the right: greater than $-1$ , and $-1$ not included.

(b) Filled circle at 4, arrow to the left: less than or equal to 4.

(d) Filled circle at $-6$ , arrow to the right: greater than or equal to $-6$ .

(a) $x > -1$ (b) $x \le 4$ (c) $x < 1.5$ (d) $x \ge -6$

Question 5

Match each inequality with its number line.
(i) $x > 2$ (ii) $x \ge -2$ (iii) $2 \ge x$ (iv) $x < -2$

ShowHide solution

(i) $x > 2$ : open circle at 2, arrow to the right. That is B.

(ii) $x \ge -2$ : filled circle at $-2$ , arrow to the right. That is A.

(iii) $2 \ge x$ is $x \le 2$ : filled circle at 2, arrow to the left. That is D.

(iv) $x < -2$ : open circle at $-2$ , arrow to the left. That is C.

(i) B (ii) A (iii) D (iv) C

Question 6

Each number line was drawn for the inequality above it, and each is wrong. Say what is wrong, and draw it correctly.

(a) $x \ge 1$

(b) $-2 > x$

ShowHide solution

(a) The circle is open, but $\ge$ includes 1. The circle must be filled.

(b) $-2 > x$ is $x < -2$ , so the arrow must go to the left.

(a) The circle should be filled. (b) The arrow should go to the left. (c) The circle should be at 1.5.

Question 7

The number line shows the solutions of an inequality in $x$ .
(a) Write down the inequality.
(b) Write down two numbers that are solutions and one number that is not.
(c) Is 3 a solution? Is 2.9 a solution?

ShowHide solution

(a) Open circle at 3, arrow to the left: $x$ is less than 3, and 3 is not included.

(b) Any number less than 3 is a solution, such as 0 or $-5$ . Any number from 3 upwards is not, such as 4.

(a) $x < 3$ (b) For example, 0 and $-5$ are solutions, and 4 is not. (c) 3 is not a solution; 2.9 is.

Question 8

A ride at a theme park is open only to children whose height is at least 120 cm. Let a child's height be $h$ cm.
(a) Write an inequality for the heights of the children who may go on the ride, and show it on a number line.
(b) May a child of height 118 cm go on the ride?

ShowHide solution

(a) "At least 120" allows 120 and anything more: a filled circle at 120, arrow to the right. The marks are every 5 cm.

(b) 118 is less than 120, so it is to the left of the circle.

(a) $h \ge 120$ , drawn above. (b) No: 118 cm is less than 120 cm.

Question 9

The temperature of a cold room must be kept below $-2$ °C. Let the temperature be $t$ °C.
(a) Write an inequality in $t$ and show it on a number line.
(b) Is a temperature of $-2$ °C allowed? Is $-5$ °C allowed?

ShowHide solution

(a) "Below $-2$ " does not allow $-2$ itself: an open circle at $-2$ , arrow to the left.

(b) $-2$ is not a solution, because the circle is open. $-5$ is to the left of $-2$ , so it is a solution.

(a) $t < -2$ , drawn above. (b) $-2$ °C is not allowed; $-5$ °C is allowed.

Question 10

The number line shows the ages, $a$ years, of the people who may watch a film rated M18.
(a) Write down the inequality shown.
(b) Say in words who may watch the film.
(c) May a person aged 17 watch it? May a person aged 18?

ShowHide solution

(a) Filled circle at 18, arrow to the right: greater than or equal to 18.

(b) The boundary value 18 is included.

(a) $a \ge 18$ (b) People aged 18 and above. (c) A person aged 17 may not; a person aged 18 may.

Key Points

An open circle at the boundary value for $<$ and $>$ : the boundary value is not a solution. A filled circle for $\le$ and $\ge$ : it is.
With the unknown on the left, the line and arrow go to the right for $>$ and $\ge$ , and to the left for $<$ and $\le$ .
If the unknown is on the right, rewrite first: $4 < x$ is $x > 4$ , and its arrow goes to the right.
A boundary value that is not a whole number, such as 2.5, goes in its place between the marks, with its value written above it.
To read a number line: the boundary value is at the circle; the arrow gives greater or less; the circle gives whether the boundary value is included.