What can we do to isolate a variable in an inequality? Graph both inequalities.
Practice makes perfect
We want to graph the numbers that are solutions of both given inequalities.
z-3 ≤ 8
6z < 72
We can do this by solving both inequalities and graphing them on a number line. Inequalities can be solved in the same way as equations — by performing inverse operations on both sides until the variable is isolated. The only difference is that when we divide or multiply by a negative number, we must reverse the inequality sign. Let's start with the first inequality!
We found that all values of z less than or equal to 11 will satisfy the first inequality. Now let’s graph the inequality on a number line. Since z can equal 11, we draw a closed circle at this point. We know that z is all values less than or equal to 11, so we will shade the part of the number line that represents numbers less than 11. This means that we shade to the left of our point at 11, including 11 with the closed circle.
Now we can solve the second inequality and graph its solution. Let's do it!
We found that all values of z less than 12 will satisfy the inequality. Now let’s graph the inequality on a number line. Since z cannot equal 12, we draw an open circle at this point. We know that z is all values less than 12, so we will shade the part of the number line that represents numbers less than 12. This means that we shade to the left of our point at 12.
We have solved both inequalities. Let's graph both solutions on the same number line. Any number in the region that is shaded by both inequalities is a solution to both inequalities. Let's see the graph!
As we can see, the numbers less than or equal to 11 are included in the solution set of both inequalities. Let's shade just this region to show the numbers that solve both inequalities. Remember to use a closed circle at 11! This means that we shade to the left of our point at 11.
