Solving and Graphing Linear Inequalities

How is graphing the number line affective in helping to illustrate solving inequalities? Essential Question: What are the different procedures in solving inequalities, compound inequalities?

What’s an inequality? Is a range of values, rather than ONE set number An algebraic relation showing that a quantity is greater than or less than another quantity. Speed limit:

Symbols Less than Greater than Less than OR EQUAL TO Greater than OR EQUAL TO

Vocabulary Inequalities: mathematical sentence built from expressions using the symbols, ≤, or ≥. Closed Circle: The number is a solution Open circle: The number is not a solution.

Solutions…. You can have a range of answers…… All real numbers less than 2 x< 2

Solutions continued… All real numbers greater than -2 x > -2

Solutions continued… All real numbers less than or equal to 1

Solutions continued… All real numbers greater than or equal to -3

Did you notice, Some of the dots were solid and some were open? Why do you think that is? If the symbol is > or < then dot is open because it can not be equal. If the symbol is  or  then the dot is solid, because it can be that point too.

Graphing Inequalities < Less Than x < 4 -3 > x

Graphing Inequalities < Less Than x < 4 -3 > x > Greater Than X > 2 -4 < x

Graphing Inequalities < Less Than x < 4 -3 > x > Greater Than X > 2 -4 < x ≤ Less than or Equal X ≤ 6-2 ≥ x

Graphing Inequalities < Less Than x < 4 -3 > x > Greater Than X > 2 -4 < x ≤ Less than or Equal X ≤ 6-2 ≥ x ≥ Greater than or Equal X ≥ ≤ x

Math Starter Graph each inequality on a number line ≥ y 2. b < -5

Write and Graph a Linear Inequality Sue ran a 2-K race in 8 minutes. Write an inequality to describe the average speeds of runners who were faster than Sue. Graph the inequality. Faster average speed > Distance Sue’s Time

Solving Inequalities EquationInequality x + 14 = 16 x + 14 < 16

Solving an Inequality Solving a linear inequality in one variable is much like solving a linear equation in one variable. Isolate the variable on one side using inverse operations. Add the same number to EACH side. x – 3 < 5 Solve using addition: +3 x < 8

Solving Using Subtraction Subtract the same number from EACH side. -6

Solving Inequalities < Less Than > Greater Than ≤ Less Than or Equal ≥ Greater Than or Equal 12 + x < 21 Open Circle 32 < x + 21 Open Circle x – 10 ≤ 9 Closed Circle 0.75 ≤ d Closed Circle

Using Subtraction… Graph the solution

Using Addition… Graph the solution.

Practice Copy problem, show steps, graph on number line 1. y – 21 > > x - 3

Practice Book page 280 numbers 18 to 23 all 36, 37, 38 Copy problem Show steps to solve Graph solution on number line

Math Starter Copy problem, solve and graph solution on number line. 1. x – 5 > ≤ a + 12

Solving using Multiplication Multiply each side by the same positive number. (2)

Solving Using Division Divide each side by the same positive number. 33

THE TRAP….. When you multiply or divide each side of an inequality by a negative number, you must reverse the inequality symbol to maintain a true statement.

Solving by multiplication of a negative # Multiply each side by the same negative number and REVERSE the inequality symbol. See the switch

Solving by dividing by a negative # Divide each side by the same negative number and reverse the inequality symbol. -2

Examples Solve and graph each inequality 1. 8x ≥ x > 70 3.K < m ≤ 18 4

Practice Book page 286 numbers 19 to 24 all Copy problem Show steps Graph solution

6. 8. x + 20 < ≥ -4x ≥ y x > ≥ 4. x – 3 ≤ – x ≥ 2

Graph on a number line

Practice Solve and graph each inequality 1. -3x + 5 > x + 8 ≥ < 3a + 12

Practice Book page 286 numbers 32 to 39 all Copy problem Show steps Graph solution

Math Starter: copy problem show steps and graph on number line 1. x + 12 > < - 12b ≤ 2c – x – 7 > 18