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.
p+3/4 < 3
1/4 p > 3/8
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 p less than 2 14 will satisfy the first inequality. Now let’s graph the inequality on a number line. Since p cannot equal 2 14, we draw an open circle at this point. We know that p is all values less than 2 14, so we will shade the part of the number line that represents numbers less than 2 14. This means that we shade to the left of our point at 2 14.
Now we can solve the second inequality and graph its solution. Let's do it!
We found that all values of p greater than 1 12 will satisfy the inequality. Now let’s graph the inequality on a number line. Since p cannot 1 12, we draw an open circle at this point. We know that p is all values greater than 1 12, so we will shade the part of the number line that represents numbers greater than 1 12. This means that we shade to the right of our point at 1 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 greater 1 12 and less than 2 14 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 open circles at both 1 12 and 2 14! This means that we have to shade starting at 1 12 with an open circle and ending at 2 14 with an open circle too.
