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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 • • • • • • • Use inverse operations −3 2𝑥 − 5 + 1 ≥ 4 −6𝑥 + 15 + 1 ≥ 4 −6𝑥 + 16 ≥ 4 −6𝑥 ≥ −12 𝑥≤2 Graph the solution 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 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: 3t + 2 < -7 or -4t + 5 < 1 • 3t < -9 or -4t < -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 |2x – 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. • 2x – 5 =13 and 2x – 5 = -13 • x = 9 or x = -4 Practice • Solve the following: • 2|x + 5| - 2 = 6 • |3x – 7| + 3 = 20 Absolute Value Equations • Since the absolute value is the distance between a number and zero, an absolute value cannot be negative. • Solve 3|2z + 9| + 12 = 10 • Subtract 12: 3|2z + 9| = -2 2 • Divide by 3: |2z + 9| = 3 • Absolute value cannot be negative so there is no solution • Be sure to check for extraneous solutions, meaning a solution which does not satisfy the original equation. • Ex: when you solve 3𝑥 + 2 = 4𝑥 + 5, you get the solutions x = -3 or x = -1. o if you substitute these into the original equation, you will find that x = -3 does not satisfy the equation and x = -1 does. So, x = -3 is extraneous and is not a solution. Inequalities • The same method can be applied to solving absolute value inequalities. • 2 2𝑥 + 4 − 3 ≥ 9 • Start by using inverse operations to isolate the absolute value expression. • 2|2𝑥 + 4| ≥ 12 add 3 • |2𝑥 + 4| ≥ 6 divide by 2 • 2𝑥 + 4 ≥ 6 𝑜𝑟 2𝑥 + 4 ≤ −6 o Notice that the symbol is reversed when you take the opposite sign. • 2𝑥 ≥ 2 𝑜𝑟 2𝑥 ≤ −10 • 𝑥 ≥ 1 𝑜𝑟 𝑥 ≤ −5 • Graph subtract 4 Assignment • Odds p.38 #27-31, 39, 43 • p.46 #43-47, 57-61