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How Do You Graph the Intersection of Two Inequalities?

Graph the intersection of x>80 and x≤105 on a number line

Summary

  1. The upside down "U" means "intersect"
  2. The intersection of two inequalities is all the values for x that make BOTH inequalities true
  3. We have an open circle and a right-pointing arrow at 80 because x is greater than 80
  4. We have a closed circle and a left-pointing arrow at 105 because x is less than or equal to 105
  5. The intersection is the area that is covered by both graphs

Notes

    1. Since 80 is one of the endpoints of our inequality, we put a circle there when we graph it
    1. The circle is OPEN because we have a greater than symbol
    1. "x is greater than 80" means that x IS NOT included in the inequality
    2. We represent this on a number line with an OPEN circle at 80
    1. To determine this, we need to look at our inequality symbol
    1. When we have a greater than symbol, we have a right-pointing arrow
    1. Since 105 is one of the endpoints of our inequality, we put a circle there when we graph it
    2. The circle is CLOSED because we have a less than or equal to symbol
    1. "x is less than or equal to 105" means that x IS included in the inequality
    2. We represent this on a number line with an CLOSED circle at 105
    1. To determine this, we need to look at our inequality symbol
    1. When we have a less than or equal to symbol, we have a left-pointing arrow
    1. The intersection is everything the two inequalities have in common
    1. If it's shaded twice, it means it is included in both inequalities and is in their intersection
    1. Any numbers less than 80 are not included in the inequality x>80
    1. When there are two shadings, that means those numbers are both greater than 80 and less than or equal to 105
    2. These are the numbers that will be in our intersection
    1. Any numbers greater than 105 are not included in the inequality x≤105
    1. These are the numbers that are in both of our inequalities
    1. Since 80 and 105 were the endpoints of our two inequalities, they are also the endpoints of our intersection
    1. Since 80 and 105 were the endpoints of our two inequalities, they are also the endpoints of our intersection
    2. We shade in between to represent the numbers included in both inequalities