1. Solving Compound Inequalities

2. Conjunction: compound inequality in the form a< x < b
Compound Inequality: is an inequality that is formed by the union (or) or intersection (and) of two simple inequalities. 2 types:
Conjunction: compound inequality in the form a< x < b
ie. -2 < x < 5
-means x value is between numbers a and b (think “betweeness”)
-graphs as starting and ending circles with line connecting them
Disjunction: compound inequality in the form x< a or x>b
ie. x < -2 or x > 5
-means x is greater than one of the numbers or less than the other (think “either/or” two separate things)
-graphs as two separate inequalities with arrows pointing away in opposite directions.

3. Learn to Read… a< x <b Read: “x is between a and b”
Read: “x is between a & b inclusive”
x < b x > a
Read: “x greater than a or x less than b”

4. Graph: 1< x < 5 Graph: 2< x < 4.5
Read as: ?
x is between 1 and 5 1 2 3 4 5 6 7
Graph: 2< x < 4.5
Read as: ?
x is between 2 and 4.5 inclusive 1 2 3 4 5 6 7

5. x greater than 2 or x less than -1
Write the inequality
x greater than 2 or x less than -1
x < -1 x > 2
Graph: -2 -1 1 2 3
*Remember:
(closed circle) • when the number is included!

6. Write the inequality in simplified form and graph:
x < 4 and x > -3 -4 -2 2 4
Because those 2 simple inequalities would have an intersection (overlap), this is a conjunction. Rewrite in simplified form as “between” the numbers that satisfy both simple inequalities (the numbers where both graphs would touch).
Place the smallest number that x is greater than on the left, place x in the center, place largest number that x is smaller than on the right.
-3 < x < 4 -4 -2 2 4

7. Solve Compound Inequalities and Graph:
Solve like a simple inequality/equation, but now you have three parts!
-9 < 6x + 3 ≤ 39
Subtract 3 from all three parts. -3 -3 -3
-12 < 6x ≤ 36
Divide each part by 6. 6 6 6
-2 < x ≤ 6
GRAPH: -3 -2 -1 1 2 3 4 5

8. Write the inequality, solve and then graph:
Eight times a number x plus 4 is between -4 and 20.
-4< 8x + 4 < 20
-8 < 8x < 16
-1 < x < 2 -2 -1 1 2 3

9. Practice: Graph 1. x > 4 and x < -2 2. -2 < x < 3 Solve -1 1 2 -2 -1 1 2 3
Solve
x > 3
3 < 2 - x < 5
x ≥ 3
-3 < x < -1
5. Write an inequality for the following graph
4≤ x ≤ 8

10. Use inequalities to solve the following problem.
What are the restriction on the value of x in the triangle? Remember that two side of a triangle must add up to be greater than the third side.
You must set up three inequalities! x 4
x + 4 > 6 x > 2
x + 6 > 4 x > -2 *length cannot be negative*
6 + 4 > x 10 > x 6
2 < x < 10