Solve the following: (8 + v)2 – 10 = 22 Sec 6.5 Solving Inequalities Using Addition and Subtraction

6-5 Solving Inequalities by Adding & Subtracting Objective The student will be able to: graph inequalities on a number line. solve inequalities using addition and subtraction.

To solve for a variable the variable must be positive with no number beside it!!  x = 10 Solved - x = -5 Not solved 2x = 5 Not solved -3x =6 Not solved

greater than or equal to INEQUALITIES Another type of sentence used in algebra is called an inequality. An inequality is used when we don't know exactly what an expression is equal to. Instead of an equals sign, we use one of these symbols: This will be easier to do if you remember that the open part of the symbol always faces the larger quantity. SYMBOL MEANING   <   less than >    greater than   ≤   less than or equal to    ≥  greater than or equal to

Graph the solution set of x < 3 When you have < or >, use an open dot!

Graph the solution set of y ≥ -5/4 • When you have ≤ or ≥, use a closed dot! Convert -5/4 to a decimal = -1.25

Graph the solution set of p ≠ 2 When you have ≠, use an open dot and shade both ways!

What is the graph of 7 ≤ m? o 7 8 6 o 7 8 6 ● 7 8 6 ● 7 8 6 Answer Now

What type of dot open or closed? < > ≤ ≥ ≠ Open Open Closed Closed Open

Addition & Subtraction property of inequality You can add or subtract the same number or term on each side of the inequality, the relationship between the two sides does not change. x - 14 < 16 y + 21 ≥ 7 x – 14 + 14 < 16 +14 y + 21 - 21 ≥ 7 - 21

1) Solve x + (-14) < 16 x - 14 < 16 + 14 + 14 x < 30 + 14 + 14 x < 30 30 + (-14) = 16 16 = 16 Solve this problem like an equation Draw “the river” Eliminate double signs Add 14 to both sides Simplify Check your answer Graph the solution o 30 31 29

2) Solve y + 21 ≥ 7 y ≥ -14 30 + (-14) = 16 16 = 16 ● - 21 -21 -14 -13 - 21 -21 y ≥ -14 30 + (-14) = 16 16 = 16 Draw the “river” Subtract 21 from both sides Simplify Check your answer Graph the solution -14 -13 -15 ●

3) Solve 8y + 3 > 9y - 14 - 8y - 8y 3 > y - 14 + 14 + 14 + 14 + 14 17 > y y < 17 8(17) + 3 = 9(17) - 14 Draw “the river” Subtract 8y from both sides Simplify Add 14 to both sides Rewrite inequality with the variable first Check your answer Graph the solution o 17 18 16

4) Solve 3r - 17 ≥ 2r + 14 - 2r - 2r r – 17 ≥ 14 + 17 + 17 r ≥ 31 ● + 17 + 17 r ≥ 31 3(31) - 17 = 2(31) + 14 Draw “the river” Subtract 2r from both sides Simplify Add 17 to both sides Check your answer Graph the solution ● 31 32 30

Solve and graph -2x + 6 ≥ 3x - 4 x ≤ 2

Joanna’s tests were 87, 93, 88 and 94 Joanna’s tests were 87, 93, 88 and 94. What must her 5th grade be to get a total of at least 459? X + 87 + 93 + 88 + 94 ≥ 459 X + 362 ≥ 459 -362 -362 X ≥ 97

solve inequalities using multiplication and division. 6-6 Solving Inequalities by Multiplying or Dividing Objective The student will be able to: solve inequalities using multiplication and division.

The inequality stays the same POSITIVE NUMBERS ONLY Multiplication & Division properties of inequalities You can multiply or divide the same positive number or positive term on each side of the inequality, the relationship between the two sides does not change. 8p ≥ -96 < -10 ≥ The inequality stays the same

The inequality gets flipped every time you * or / by a negative number NEGATIVE NUMBERS ONLY Multiplication & Division properties of inequalities When multiplying or dividing the same negative number or negative term on each side of the inequality, you must reverse the inequality sign -8p ≥ -96 > -10 ≤ < -3 The inequality gets flipped every time you * or / by a negative number

To solve for a variable the variable must be positive with no number beside it!!  x = 10 Solved - x = -5 Not solved 2x = 5 Not solved -3x =6 Not solved

1) Solve > 13 k < -52 o -52 -51 -53 Draw “the river” Clear the fraction - Multiply both sides by -4 NEW STEP!! When multiplying BOTH sides by a NEGATIVE number, SWITCH the inequality! Simplify Check your answer Graph the solution k < -52 o -52 -51 -53

2) When solving < -10 will the inequality switch? Yes! No! I still don’t know! Answer Now

2) Solve < -10 x < -30 o -30 -29 -31 Draw “the river” Clear the fraction - Multiply both sides by 3. Do you switch the inequality? No - Both sides are being multiplied by a positive number Simplify Check your answer Graph the solution x < -30 o -30 -29 -31

3) When solving will the inequality switch? Yes! No! I still don’t know! Answer Now

3) Solve a > -24 o -24 -23 -25 Draw “the river” Clear the fraction - Multiply both sides by -4. Do you switch the inequality? Yes - Both sides are being multiplied by a negative number Simplify Check your answer Graph the solution a > -24 o -24 -23 -25

4) Solve -8p ≥ -96 p ≤ 12 -8(12) = -96 ● 12 13 11 Draw “the river” Divide both sides by -8. Do you switch the inequality? Yes - Both sides are being divided by a negative number Simplify Check your answer Graph the solution p ≤ 12 -8(12) = -96 ● 12 13 11

1) Which graph represents the correct answer to > 1 -4 -3 -5 Multiply both sides by -4 Switch inequality sign K < -4 o -4 -3 -5 ● -4 -3 -5 ● -4 -3 -5 Answer Now

● ● 5) Solve 7v < -105 o o -15 -14 -16 -15 -14 -16 -15 -14 -16 -15 Divide both sides by 7 V < -15 o -15 -14 -16 ● -15 -14 -16 ● -15 -14 Answer Now

 Homework Practice Sheet 6-5 all (9-14 show work) Practice Sheet 6-6 all (show work for all) 

Homework Pg. 284 #8-26 Even 8-18 show work, graphing, and solution (3pts) 20-22, 26 write inequality solution, and the graph(3 pts) 24 just graph (1pt)