Inequalities and their Graphs

Inequalities and their Graphs What is an inequality? A statement that two expressions are not equal

Inequalities and their Graphs less than greater than < > Open circles

< = ≤ > = ≥ Inequalities and their Graphs less than or equal to greater than or equal to < = ≤ > = ≥ Closed circles

Inequalities and their Graphs X is greater than 3 Open circle Arrow points right -1 1 2 3 4 5

Inequalities and their Graphs X is less than -2 Open circle Arrow points left -6 -5 -4 -3 -2 -1

Inequalities and their Graphs n is greater than or equal to 8 Closed circle Arrow points right 5 6 7 8 9 10 11

Inequalities and their Graphs p is less than or equal to 2 Closed circle Arrow points left -2 -1 1 2 3 4

x ≥ 1 x > -3 x < 2 x ≤ 0 Name the Graph: -3 -2 -1 1 2 3 -3 -2 -1 1 2 3 x > -3 -3 -2 -1 1 2 3 x < 2 -3 -2 -1 1 2 3 x ≤ 0 -3 -2 -1 1 2 3

- 1 -1 Solving an inequality x + 1 ≤ 3 x ≤ 2 0 + 1 ≤ 3 1 ≤ 3  -3 -2 - 1 -1 x ≤ 2 -3 -2 -1 1 2 3 Check: Choose a point included on the graph and see if its true….. Choose x = 0, is x + 1 ≤ 3 true? 0 + 1 ≤ 3 1 ≤ 3 

+4 +4 Solving an inequality x - 4 ≥ -1 x ≥ 3 4 - 4 ≥ -1 0 ≥ -1  -1 1 +4 +4 x ≥ 3 -1 1 2 3 4 5 Check: Choose a point included on the graph and see if its true….. Choose x = 4, is x - 4 ≥ -1 true? 4 - 4 ≥ -1 0 ≥ -1 

-3 -3 Solving an inequality x + 3 > 2 x > -1 0 + 3 > 2 -3 -3 x > -1 -3 -2 -1 1 2 3 Check: Choose a point included on the graph and see if its true….. Choose x = 0, is x + 3 > 2 true? 0 + 3 > 2 3 > 2 

+4 +4 Solving an inequality x - 4 < -3 x < 1 0 – 4 < -3 +4 +4 x < 1 -3 -2 -1 1 2 3 Check: Choose a point included on the graph and see if its true….. Choose x = 0, is x - 4 < -3 true? 0 – 4 < -3 -4 < -3 

Solving an inequality 4x < 8 4 4 x < 2 4(0) < 8 0 < 8  -3 4 4 x < 2 -3 -2 -1 1 2 3 Check: Choose a point included on the graph and see if its true….. Choose x = 0, is 4x < 8 true? 4(0) < 8 0 < 8 

Solving an inequality x 2 ∙ > 1 ∙ 2 2 x > 2 4/2 > 1 2 > 1 > 1 ∙ 2 2 x > 2 -1 1 2 3 4 5 Check: Choose a point included on the graph and see if its true….. Choose x = 4, is x/2 > 1 true? 4/2 > 1 2 > 1 

Warning: dividing by a negative number may be different…. Solving an inequality Warning: dividing by a negative number may be different…. -3x ≤ 6 -3 -3 x ≤ -2 -3 -2 -1 1 2 3 Check: Choose a point included on the graph and see if its true….. Choose x = -3, is -3x ≤ 6 true? -3(-3) ≤ 6 NO 9 ≤ 6

Solving an inequality -3x ≤ 6 -3 -3 x ≥ -2 -3(0) ≤ 6 0 ≤ 6  -3 -2 -1 Rule: When dividing (or multiplying) by a negative number, you need to switch the sign in the opposite direction -3x ≤ 6 -3 -3 x ≥ -2 -3 -2 -1 1 2 3 Check: Choose a point included on the graph and see if its true….. Choose x = 0, is -3x ≤ 6 true? -3(0) ≤ 6 0 ≤ 6 

Solving an inequality x -2 ∙ > 1 ∙ -2 -2 x < -2 -4/-2 > 1 Rule: When dividing (or multiplying) by a negative number, you need to switch the sign in the opposite direction -2 ∙ > 1 ∙ -2 -2 x < -2 -6 -5 -4 -3 -2 -1 Check: Choose a point included on the graph and see if its true….. Choose x = -4, is x/-2 > 1 true? -4/-2 > 1 2 > 1 