1 5 Solving Inequalities Solve and graph inequalities

1 -5 Solving Inequalities Solve and graph inequalities by using properties of inequalities.

Graphing Inequalities • Open dot for < or > • Closed dot for ≥ or ≤ • If the inequality symbol is open toward the variable, shade to the right. • If the inequality symbol is pointed toward the variable, shade to the left.

Writing an Inequality from a Sentence • What inequality represents “five fewer than a number is at least 12”? • What numbers are we dealing with? • Any variables? • What words indicate an operation? • What words indicate an inequality? • X – 5 ≥ 12

Properties of Inequalities • If you multiply or divide both sides of an inequality by a negative number, the symbol flips.

Solving •

No Solution or All Real Numbers • There is no solution if all variables cancel and the statement is false. o Ex: -2 > 7 (no variables and we know that -2 is not greater than 7) • All real numbers are solutions if all variables cancel and the statement is true. o Ex: -15 ≤ 8 (no variables and it is true that -15 is less than 8) o Same as infinitely many solutions

Compound Inequality • Consists of two distinct inequalities joined by the word and or the word or • You can find the solutions by identifying where the solutions overlap or by combining the solutions to form a larger solution set.

Using the word “And” • Contains the overlap of the graphs of two inequalities that form a compound inequality. • EX: x ≥ 3 and x ≤ 7 • Can also be written 3 ≤ x ≤ 7 o This is only for a compound inequality using the word “and”

Using the word “Or” • Contains each graph of the two inequalities that form the compound inequality. • Used when there is no overlap. • EX: x < -2 or x ≥ 1

Writing a Compound Inequality • All numbers that are greater than -2 and less than 6 • Key information • n > -2 and n < 6 • -2 < n < 6 • Graph

Solving • A solution to a compound inequality involving and is any number that makes both inequalities true. • EX: -3 ≤ m – 4 < -1 • Isolate the variable by adding 4 to each piece • -3 + 4 ≤ m – 4 + 4 < -1 + 4 • 1≤m<3

Solving • A solution to a compound inequality involving or is any number that makes either inequality true. • You must solve each inequality separately. • EX: 3 t + 2 < -7 or -4 t + 5 < 1 • 3 t < -9 or -4 t < -4 • t < -3 or t > 1

1 -6 Absolute Value Equations and Inequalities Write and solve absolute value equations and inequalities by applying the definition of absolute value.

Solving an Absolute Value Equation • What are the solutions of |x| + 2 = 9 • Solve using inverse operations • |x| = 7 so… what is x? • x = 7 or x = -7 • Why?

Key Concept • What about |2 x – 5| = 13? • To solve an equation in the form |A| = b, where A represents a variable expression, solve both A = b and A = -b. • When solving, always isolate the absolute value expression first. Do not use inverse operations on what is inside. • 2 x – 5 =13 and 2 x – 5 = -13 • x = 9 or x = -4

Practice • Solve the following: • 2|x + 5| - 2 = 6 • |3 x – 7| + 3 = 20

Absolute Value Equations •

Inequalities •

Assignment • Odds p. 38 #27 -31, 39, 43 • p. 46 #43 -47, 57 -61