Graph Absolute Value Inequalities on a Number Line

Graph Inequalities on a Number Line - How it Works - Video

Example 1 Part 1

Example 1 Part 1:

When dealing with inequalities that have absolute value, we have two cases. Remember absolute value is a distance, we can either go left 3 steps (negative) or right 3 steps (positive). Since absolute value is already solve for, we can solve for the first case.

| x - 5 | < 3

x - 5 < 3

x < 8

Rewrite the inequality

Drop absolute value signs

Add 5 to both sides

For the second case we have to flip the inequality. We must flip the sign and we must flip the number.

So we flipped the sign from greater than to less than, and we flipped the number from 3 to -3.

| x - 5 | > -3

x - 5 > -3

x > 2

Rewrite the inequality

Drop absolute value signs

Add 5 to both sides

Finally since we have an and statement, we combine the two inequalities to make one. We always put the smaller number first in this case 2. X goes in the middle, and the biggest number is last. Just make sure that the signs go in the same direction. So our compound inequality is 2 < x < 8.

Example 1 Part 2

Example 1 Part 2:

Now, we must graph. Let's start with the smaller number, 2.

Since 4 is greater than 2, we draw an arrow to the right, and everything to the right makes this part of the inequality true, but we have to be careful.

Example 1 Part 3

Example 1 Part 3:

Now, let's finish the graph. Let's finish

Since 0 is less than 8, we draw an arrow to the left, and everything to the left makes this part of the inequality true, but we have to be careful.

Example 1 Part 4

Example 1 Part 4:

Since we have an and statement, we need both of the inequalities to be true at the same time. So we only shade the numbers between 2 and 8 but not including 2 or 8 to represent the graph.

Finally our compound inequality is our domain or our input values. You also might see it written in interval notation, (2, 8). Since we have less than or less than or equal to we put parenthesis around each number.

Example 2 Part 1

Example 2 Part 1:

When dealing with inequalities that have absolute value, we have two cases. Remember absolute value is a distance, we can either go left 2 steps (negative) or right 2 steps (positive). Since absolute value is already solve for, we can solve for the first case.

| x - 24 | 2

x - 24 2

x 26

Rewrite the inequality

Drop absolute value signs

Add 24 to both sides

For the second case we have to flip the inequality. We must flip the sign and we must flip the number.

So we flipped the sign from greater than or equal to to less than or equal to, and we flipped the number from 2 to -2.

| x - 24 | -2

x - 24 -2

x 22

Rewrite the inequality

Drop absolute value signs

Add 24 to both sides

Finally since we have an or statement, we don't need to combine the two inequalities. But, we are going to put the smaller number first in this case 22. And the biggest number second in this case, 26. Just make sure that the signs go in the same direction. So our compound inequality is x 22 or 26 x.

Example 2 Part 2

Example 2 Part 2:

Now, we must graph. Let's start with the smaller number, 22.

Since 20 is less than or equal to 22, we draw an arrow to the left, and everything to the left makes this part of the inequality true.

Example 2 Part 3

Example 2 Part 3:

Now, let's finish the graph. Let's finish

Since 28 is greater than or equal to 26, we draw an arrow to the right, and everything to the right makes this part of the inequality true.

Example 2 Part 4

Example 2 Part 4:

Since we have an or statement, we only need one of the inequalities to be true at one time. So we shade both inequalities for our graph .

Finally our compound inequality is our domain or our input values. You also might see it written in interval notation, (-∞, 22] U [26, +∞). Since we have less than or less than or equal to we put brackets around each number.

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