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1.5 B) Inequalities on a Number Line – Part 1
1.5 B) Inequalities on a Number Line – Part 1
We are going to be looking at drawing inequalities on a number line in this section. An example of an inequality that we are going to draw is:
When we are showing inequalities on a number line, we use circles to show the end values. There are two different types of circles for inequalities on a number line. These two different types are:
We then have a line going from the circles to show the possible values that the unknown may be. For the example f > 5, the line will be showing the possible values for f .
This will all make a lot more sense after we have had a look at a few examples.
- Open circles – used to show that a number cannot be a possible value; we use open circles when we have less than (<) or greater than (>).
- Solid (or closed) circles – used to show numbers that could be a possible value; we use solid circles for less than or equal to (≤) and for greater than or equal to (≥).
We then have a line going from the circles to show the possible values that the unknown may be. For the example f > 5, the line will be showing the possible values for f .
This will all make a lot more sense after we have had a look at a few examples.
Example 1
Draw the inequality f > 5 on the number line below.
Draw the inequality f > 5 on the number line below.
This inequality states that f is a value that is greater than 5. The inequality is just greater than and does not include “or equal to”. This means that we will be using an open circle. We therefore draw an open circle at 5. This is shown on the number line below:
The inequality tells us that f is greater than 5. This means that f can be any values on the right of the circle that we have drawn at 5. We represent this by drawing a rightwards arrow going from the circle at 5. The answer is shown below:
Example 2
Draw the inequality x ≤ 12 on the number line below.
Draw the inequality x ≤ 12 on the number line below.
The inequality in the question is ≤, which means less than or equal to. As there is an “or equal to” in the inequality, it means that we will be using a solid circle. Therefore, the first step in answering this question is to draw a solid circle at 12.
The inequality tells us that x is less than or equal to 12. This means that x can be any values that are on the left of the solid circle. We represent this by drawing a leftwards arrow going from the circle at 12. The answer is shown below:
Example 3
Draw the inequality 3 < g ≤ 8 on the number line below.
Draw the inequality 3 < g ≤ 8 on the number line below.
This question is a little bit trickier because there are two different parts to the inequality. The different parts tell us that g is greater than 3, and g is less than or equal to 8. The first step in answering a question like this is to draw circles at the key values in the inequality. When we are drawing circles, we need to check whether the circles will be open or closed; we do open circles when the value is not equal to and closed circles when the value is equal to. Therefore, the circle at 3 is going to be open because g is greater than 3 (there is no “or equal to”) and the circle at 8 is going to be solid because g is less than or equal to 8. These circles are shown on the number line below.
The inequality tells us that g can be values that are in between 3 and 8. Therefore, we draw a line from the circle at 3 to the circle at 8. The final answer is shown below.
Example 4
Draw the inequalities 6 ≥ y and 12 < y on the number line below.
Draw the inequalities 6 ≥ y and 12 < y on the number line below.
There are two different inequalities in this question, and we answer this type of question by drawing the inequalities out seperately. I am going to start by drawing the first inequality, which is 6 ≥ y. This inequality is saying that 6 is greater than or equal to y. As there is an “or equal to” involved, it means that we will have a solid circle at 6. The inequality tells us that 6 is greater than or equal to y. An alternative way of viewing this is that y is less than or equal to 6. This means that we will have a leftwards arrow going from the circle at 6. This inequality has been drawn on the number line below.
The next step is to draw the other inequality; 12 < y, which states that 12 is less than y. As there is no “or equals to”, the circle at 12 will be open. The inequality states that 12 is less than y. An alternative way of viewing this is that y is greater than 12. This means that we will have an arrow going rightwards from the circle at 12.
We now have both of the inequalities drawn on the above number line.
I usually find it easier to work with inequalities when we have the unknown first. That is why I viewed both of the inequalities as y is greater than or less than a number, rather than a number is greater than or less than something. You may find it easier to do the same.
I usually find it easier to work with inequalities when we have the unknown first. That is why I viewed both of the inequalities as y is greater than or less than a number, rather than a number is greater than or less than something. You may find it easier to do the same.