What does it mean when the circles are open or closed? Where is the graph shaded?
-3< x< 6
Practice makes perfect
Let's first look at where the shaded portion of the graph is. When a graph is shaded between two points, it represents an "and" compound inequality. This is because the value of the variable must be greater than (or greater than or equal to) the lesser point and less than (or less than or equal to) the greater point.
Let's call the variable this compound inequality represents x and consider what inequalities could describe its value.
Lesser Point
The graph is shaded to the right of -3, and the circle is open, so we can say that the value of x is greater than -3.
x> -3
Greater Point
The graph is also shaded to the left of 6 and the circle is open. This tells us that x is less than 6.
x< 6
Compound Inequality
Notice that the solution set is "sandwiched" between the two points. This tells us that we have an "and" compound inequality. Rearranging x> -3 will allow us to visualize this "sandwich" when we write the compound inequality algebraically.
x> -3 ⇔ -3< x
Combining these two individual inequalities gives us a compound inequality: -8 is less than x and x is less than -3.
-3 < x and x < 6 ⇔ -3< x< 6
