Addition Property of Inequalities If the same number is added to each side of a true inequality, the resulting inequality is also true. For all numbers a, b, and c, the following are true. – 1) If a > b, then a + c > b + c – 2) If a < b, then a + c < b + c

Addition Property of Inequality x – 12 > 8Original Inequality x – > Add 12 to each side x > 20 Simplifying Try These: 1) 22 > m – 82) d – 14 > -19

Subtraction Property of Inequalities If the same number is subtracted from each side of a true inequality, the resulting inequality is also true. For all numbers a, b, and c, the following are true. – 1) If a > b, then a - c > b - c – 2) If a < b, then a - c < b - c

M + 19 > 56Original Inequality M + 19 – 19 > 56 – 19Subtract 19 from both sides M > 37Simplify – 1) x + 23 < 142) p + 8 < 18 Subtraction Property of Inequalities

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Linear Inequalities in One Variable Multiplication Property of Inequality

Of the students surveyed at Madison High School, fewer than eighty-four said they have never purchased an item online. This is about one eighth of those surveyed. How many students were surveyed? – 1n/8 < 84Original Inequality – (8) 1n/8 < 84(8)Multiply each side by 8 – n < 672Simplify

Multiplication Property of Inequality Examples: 1.) - 3r/7 -10

Division Property of Inequalities If both sides of a true inequality are divided by a positive number, the resulting inequality is also true. For any real numbers a and b and any positive real number a, – If a > b, then a/c > b/c – If a < b, then a/c < b/c

Division Property of Inequalities If both sides of a true inequality are divided by a negative number, the direction of the inequality sign is reversed to make the resulting inequality true. For any real numbers a and b and any negative real number c, – If a > b, then a/c < b/c – If a b/c

Distributive Property Inequality ExampleSteps Done 4(3t – 5) + 7 > 8t + 3 Original Inequality 12t – > 8t + 3Distributive Property 12t – 13 > 8t + 3Combine Like terms 4t – 13 > 3Subtract 8t from each side 4t > 16Add 13 to each side 4t/4 > 16/4Divide 4 from both sides T > 4Simplify

Distributive Property 1) 6(5z – 3) -3(8-h)

Empty Set and All Reals 9t – 5(t-5) < 4(t-3) 9t – 5t + 25 < 4t – 12Distributive Property 4t + 25 < 4t – 12 Combine Like Terms 4t + 25 – 4t < 4t – 12 – 4t Subtract 4t from both sides 25 < 12Simplify If the inequality is not True then the answer is NO SOLUTION!

Empty Set and All Reals 3(4m + 6) < (2m -4) 12m + 18 < m – 24Distributive 12m + 18 < 12m + 18Combine Like Terms 12m+18–12m < 12m + 18 – 12m Subtract 12m 18 < 18Simplify

Empty Set and All Reals 1) 18 – 3(8c + 4) > -6(4c – 1) 2) 46 < 8m – 4(2m + 5)

Multi-Step Inequality Write and solve an inequality to find the sales Mrs. Jones needs if she earns a monthly salary of $2,000 plus a 10% commission on her sales. Her goal is to make at least $4,000 per month. What sales does she need to make her goal?

Write and Solve the Inequality Five minus 6 times a number is more than four times the number plus 45. Two more than half a number is greater than twenty-seven

Solve and Graph Maggie has scores of 98, 86, and 88 on her first three tests in algebra. If she wants an average of at least 90 after her fourth test, what score must she make on that test?

Problems to be Done! 2 – Addition 2 – Subtraction 2 – Multiplication 2 – Distributive 2 – Empty Set 1 – Multi-Step Word Problem 2 – Write and Solve 3 – Solve and Graph

Answers Addition: 1) m -5 Subtraction: 1) x < -92) p < 10 Multiplication: 1) r > -492) p < 7 and ½ Distributive: 1) z > -3 2) h < 36 Empty Set: 1) Real Numbers2) No solution Multi-Step 1) x > 20,000 Write & Solve: 1) n 50 Solve and Graph: 1) x 7/3 – 3) x > 88