Solving Compound Inequalities

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.

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”

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

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!

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

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 6 7

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

Practice: Graph 1. x > 4 and x < -2 2. -2 < x < 3 Solve -3 -2 -1 1 2 3 4 5 -2 -1 1 2 3 Solve 2 + x > 3 3 < 2 - x < 5 x ≥ 3 -3 < x < -1 5. Write an inequality for the following graph 2 3 4 5 6 7 8 9 10 4≤ x ≤ 8

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