Notes 8th Grade Math McDowell Chapter 2

Graphing and Writing Inequalities 1/5 Inequality signs < > £ ³ Open dot Closed dot less than fewer than greater than more than exceeds less than or equal to no more than at most greater than or equal to no less than at least

Order Matters This chart only works if the variable comes first: x > 4 x is greater than 4 The variable comes first and we can use the chart to graph 4 > x 4 is greater than x The variable does not come first Flip the whole inequality around x < 4 is equivalent

Solution of An Inequality Any value that makes the inequality true There can be infinity many solutions Which are solutions of x > -3? 5, 9, -2, or –37 Replace x with the possible solution and determine if it is true.

≤ Has a closed circle and points left Graphing Inequalities Graph c > -2. > Has an open circle and points right Graph m ≤ 4. ≤ Has a closed circle and points left

Writing Inequalities from the graph Write an inequality from each graph Closed circle and points right ≤ Circle is over the 1 x ≤ 1 Open circle and points left > Circle is over the -4 x > -4.

You Try Workbook p 35-36 # all

Solving Inequalities by Adding or Subtracting 1/6 The Inequality sign When adding or subtracting, treat the inequality sign just like you would an = sign Perform the inverse to isolate the variable and leave the inequality sign alone.

Ask three questions about the inequality: Reminder Ask three questions about the inequality: 1. What is the variable? 2. What operation is performed on the variable? 3. What is the inverse operation? (The one that will undo what is being done to the variable)

Solve x + 7 > 12 Example x Addition. Subtraction. x + 7 > 12 - 7 1. What is the variable? 2. What operation is being performed on the variable? Addition. 3. What is the inverse operation (the one that will undo what is being done to the variable)? Subtraction. The subtraction property of inequality tells us to subtract the same thing on both sides to keep the equation equal. x + 7 > 12 - 7 - 7 x > 5

Example y Subtraction. Addition. y – 10 ≤ -12 +10 +10 y ≤ -2 Solve y – 10 ≤ -12 Example y 1. What is the variable? 2. What operation is being performed on the variable? Subtraction. 3. What is the inverse operation (the one that will undo what is being done to the variable)? Addition. The addition property of inequality tells us to add the same thing on both sides to keep the equation equal. y – 10 ≤ -12 +10 +10 y ≤ -2

You Try Workbook p 37-38 # all

Solving Inequalities by Multiplying or Dividing 1/11 When multiplying or dividing on both sides of an inequality sign watch for negatives. The Inequality sign When multiplying or dividing by a positive number leave the inequality sign alone. When multiplying or dividing by a negative number, flip the inequality sign.

Solve 2a < 10 Example a Multiply by 2 Divide by 2 2a < 10 2 2 1. What is the variable? 2. What operation is being performed on the variable? Multiply by 2 3. What is the inverse operation (the one that will undo what is being done to the variable)? Divide by 2 The division property of inequality tells us to divide the same thing on both sides to keep the equation equal. 2a < 10 2 is positive so leave the inequality sign alone. 2 2 a < 5

Solve m/-4  3 Example m Divide by -4 Multiply by -4 m/-4  3 X-4 X-4 1. What is the variable? 2. What operation is being performed on the variable? Divide by -4 3. What is the inverse operation (the one that will undo what is being done to the variable)? Multiply by -4 The division property of inequality tells us to divide the same thing on both sides to keep the equation equal. m/-4  3 -4 is negative so flip the inequality sign. X-4 X-4 m  -12

You Try Workbook p 39-40 # all

Solving Two-Step Equations 1/19 Two operations are required to isolate x Plug and Chug to check your answer

Ask five questions about the inequality: Solving Two Step Equations 1. What is the variable? 2. What is the first operation being done to the variable? 3. What is the inverse operation? 4. What is the second operation being done to the variable? 5. What is the inverse operation?

Work Backwards First—undo any addition or subtraction Second—undo any multiplication or division

Example -2x – 5 = 25 = 30 x = -15 -2(-15) – 5 = 25 30 – 5 = 25 25 = 25 Draw “the road” Example Work Backwards -2x – 5 = 25 Add 5 to both sides + 5 + 5 Simplify x • -2  -2 -5 +5 -2x = 30 Divide both sides by -2 -2 -2 Simplify x = -15 -2(-15) – 5 = 25 Check your answer Plug and Chug 30 – 5 = 25 25 = 25

You Try Solve y – 4 = -12 2 2. –4m + 10 = 30 3. ¼a – 13 = -3

You Try Workbook p 113