Inequalities on a Number Line

A filled circle means that number is included.

An open circle means that number is not included.

Decide whether 5 is included, then which way the values go.

Check whether the circle is open or filled before you choose.

Include a number with a filled circle, but not a number with an open circle.

x has to be a whole number that is bigger than 2.5.

1. x must be greater than 2.5, and 2.5 is not an integer anyway.
2. The integers greater than 2.5 are 3, 4, 5, …
3. The smallest possible value of x is 3.

≤ means the number is included; < means it is not.

1. -3\leq n means -3 is included.
2. n<2 means 2 is not included.
3. The integers are -3, -2, -1, 0 and 1.

Both circles are filled, so both end numbers are included.

1. The filled circle at -2 means -2 is included: -2\leq x.
2. The filled circle at 3 means 3 is included: x\leq3.
3. The inequality is -2\leq x\leq3.

Look at each circle separately: open means not included, filled means included.

1. The open circle at -4 means -4 is not included: -4<x.
2. The filled circle at 1 means 1 is included: x\leq1.
3. The inequality is -4<x\leq1.

Decide which end is included and which is not.

1. -1<x: -1 is not included, so there is an open circle at -1.
2. x\leq4: 4 is included, so there is a filled circle at 4.
3. The values are between them, so the circles are joined by a line.

“At least 4” means 4 is allowed. “Less than 9” means 9 is not allowed.

Part (a)

1. At least 4 means n\geq4, which is the same as 4\leq n.
2. Less than 9 means n<9.
3. So 4\leq n<9. On a number line: a filled circle at 4 and an open circle at 9.

Part (b)

1. n<9, and a number of people must be a whole number.
2. The greatest number of people is 8.

For each end of the inequality, decide whether the number is included.

1. -4\leq x means -4 is included, so the circle at -4 should be filled. Omar's is open.
2. x<1 means 1 is not included, so the circle at 1 should be open. Omar's is filled.
3. Omar's line starts and stops at the right numbers, so the two mistakes are the circles.

Neither −7 nor 5 is included. List the integers in between and count them.

1. -7 and 5 are not included.
2. The integers are -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4.
3. Remember to count 0. There are 11 possible values.

The labels are whole numbers, but there is a mark halfway between each pair.

1. Each small step on this number line is 0.5.
2. The circle is halfway between 1 and 2, so it is at 1.5.
3. It is open, so 1.5 is not included, and the arrow points right.
4. The inequality is x>1.5.

Put the smaller number on the left and the larger number on the right, keeping each sign.

1. x>-2 is the same as -2<x.
2. Put x\leq5 on the other side.
3. So -2<x\leq5.

Write down the inequality each number line shows, then find the integers that are in both.

1. Number line A: open circle at -3, arrow right, so x>-3.
2. Number line B: filled circle at 2, arrow left, so x\leq2.
3. Both together: -3<x\leq2.
4. The integers are -2, -1, 0, 1 and 2.

List all the integers first, then keep the even ones. Negative numbers and 0 can be even.

1. The integers with -5\leq n<7 are -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6.
2. The even ones are -4, -2, 0, 2, 4 and 6.
3. 0 is even, and 7 is not included.

List the possible values first. Pairs like −1 and 1 add to 0.

1. The filled circle at -4 means -4 is included. The open circle at 3 means 3 is not.
2. So -4\leq n<3, and the values are -4, -3, -2, -1, 0, 1, 2.
3. -2 and 2 add to 0, and so do -1 and 1.
4. That leaves -4+(-3)+0=-7.

Both ends are negative numbers. Read each circle carefully.

Part (a)

1. The open circle at -6 means -6 is not included: -6<x.
2. The filled circle at -2 means -2 is included: x\leq-2.
3. The inequality is -6<x\leq-2.

Part (b)

1. -6 is not included and -2 is included.
2. The integers are -5, -4, -3 and -2.

4 is not included. List the possible values of x, starting with the largest, until you have six.

1. x<4, so the largest value of x is 3.
2. Six integers counting down from 3: 3, 2, 1, 0, -1, -2.
3. a\leq x means a itself is included, so the smallest value, -2, is a.
4. a=-2.