a Does the solution set in the given graph lie to the left or right of the endpoint? What does the closed circle indicate?
B
b Does the solution set in the given graph lie to the left or right of the endpoint? What does the open circle indicate?
C
c What does it mean when the circles are open or closed?
D
d What does it mean when the circles are open or closed?
A
a x ≤ 6
B
b x > 1
C
c 2 ≤ x < 7
D
d - 3 ≤ x ≤ - 1
Practice makes perfect
a Before we can write an inequality for the given graph, we need to take note of the endpoint and the direction of the inequality.
First, we see that the circle at the endpoint is closed.
This means that the inequality is non-strict and 6 is included in the solution set.
Next, we observe that the solution set lies to the left of the endpoint.
Using x to represent the solution set, we can say that any value of x is less than or equal to 6.
x ≤ 6
b Before we can write an inequality for the given graph, we need to take note of the endpoint and the direction of the inequality.
First, we see that the circle at the endpoint is open.
This means that the inequality is strict and 1 is not included in the solution set.
Next, we observe that the solution set lies to the right of the endpoint.
Using x to represent the solution set, we can say that any value of x is greater than 1.
x > 1
c 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 2, and the circle is closed, so we can say that the value of x is greater than or equal to 2.
x ≥ 2
Greater Point
The graph is also shaded to the left of 7 and the circle is open. This tells us that x is less than 7.
x< 7
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 ≥ 2 will allow us to visualize this "sandwich" when we write the compound inequality algebraically.
x ≥ 2 ⇔ 2 ≤ x
Combining these two individual inequalities gives us a compound inequality: 2 is less than or equal to x and x is less than 7.
2 ≤ x and x < 7 ⇔ 2≤ x< 7
d Let's first look at where the shaded portion of the graph is. When a graph is shaded between two points, it represents an andcompound 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 closed, so we can say that the value of x is greater than or equal to - 3.
x ≥ - 3
Greater Point
The graph is also shaded to the left of - 1 and the circle is open. This tells us that x is less than or equal to - 1.
x ≤ - 1
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: - 3 is less than or equal to x and x is less than or equal to - 1.
- 3 ≤ x and x ≤ - 1 ⇔ - 3≤ x≤ - 1
