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1.5 C) Inequalities on a Number Line – Part 2
1.5 C) Inequalities on a Number Line – Part 2
In the previous section, we were given an inequality and asked to draw it on a number line. In this section, we are going to be given a number line with an inequality on it and asked to write the inequality. The key thing to remember when we are finding an inequality is that an open circle means that the inequality does not include “or equal to” and a solid circle means that the inequality does include “or equal to”.
Example 1
The number line below is for the variable a. Write the inequality for a.
The number line below is for the variable a. Write the inequality for a.
The first step in answering this question is to look at the circle on the above number line. The circle is an open circle, which means that our inequality will not include “or equal to”; the inequality will be < or >. We now look where the line is going. The line on the above number line goes rightwards from 3. This means that a is greater than 3, thus meaning that the inequality is a > 3.
Example 2
The number line below is for the variable h. Write the inequality for h.
The number line below is for the variable h. Write the inequality for h.
There are two circles on the above number line, which means that our inequality will contain two signs. Let’s start with the first circle on the left, which is the circle at 2. This is a solid circle, which means that h could equal to 2. The line is towards the right of 2 and this means that h can be values that are greater than or equal to 2. We can write this as 2 ≤ h.
We now move onto the next circle, which is the circle at 4. The circle at 4 is open, which means that h cannot be equal to 4. The line is on the left of 4, which means that h is less than 4 and we can write the inequality as h < 4.
The final step is to check whether the inequalities can be combined. The two inequalities can be combined because h is greater than or equal to 2 and less than 4. The combined inequality is:
We now move onto the next circle, which is the circle at 4. The circle at 4 is open, which means that h cannot be equal to 4. The line is on the left of 4, which means that h is less than 4 and we can write the inequality as h < 4.
The final step is to check whether the inequalities can be combined. The two inequalities can be combined because h is greater than or equal to 2 and less than 4. The combined inequality is:
2 ≤ h < 4
We are always able to combine inequalities when two circles are connected to one another.
Example 3
The number line below is for the variable i. Write the inequality for i.
The number line below is for the variable i. Write the inequality for i.
Like the previous question, there are two circles on the number line. Therefore, the best way to answer this question is to work with each of the circles seperately.
The circle at -1 is open, which means that the variable i is not equal to -1. The line is on the left of -1, which means that i is less than -1. We can write this inequality as i < -1.
The circle at 5 is solid, which means that i could be equal to 5. The line is on the right of 5 and this means that i can be values that are greater than or equal to 5. We can write this inequality as i ≥ 5.
On the number line, the line does not connect the two circles. This means that we are unable to combine the inequalities and the answer for this question is two separate inequalities. The answer is:
The circle at -1 is open, which means that the variable i is not equal to -1. The line is on the left of -1, which means that i is less than -1. We can write this inequality as i < -1.
The circle at 5 is solid, which means that i could be equal to 5. The line is on the right of 5 and this means that i can be values that are greater than or equal to 5. We can write this inequality as i ≥ 5.
On the number line, the line does not connect the two circles. This means that we are unable to combine the inequalities and the answer for this question is two separate inequalities. The answer is:
i < -1 or i ≥ 5
We have to write the two inequalities seperately because there will not be a number that satisfies both of the inequalities. For example, -4 is less than -1, which means that the first inequality is satisfied (i < -1), but -4 is not greater than or equal to 5, which means that the second inequality is not satisfied (i ≥ 5). Therefore, we need to use two inequalities to describe the variable i.