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I can write and solve absolute-value equations and inequalities. Do Now (Turn on laptop to my calendar) Write the inequality used to solve. Success Criteria: I can interpret complicated expressions by viewing one or more of their parts as a single entity Be able to create equations and inequalities in one variable and use them to solve problems Today 1. Do Now 2. Check HW #6 3. Lesson 1.6 4. HW #7 5. Complete iReady Recall that the absolute value of a number x, written |x|, is the distance from x to zero on the number line. Because absolute value represents distance without regard to direction, the absolute value of any real number is nonnegative. Absolute-value equations and inequalities can be represented by compound statements. Consider the equation |x| = 3. The solutions of |x| = 3 are the two points that are 3 units from zero. The solution is a disjunction: x = –3 or x = 3. Example 2A: Solving Absolute-Value Equations Solve the equation. |–3 + k| = 10 –3 + k = 10 or –3 + k = –10 k = 13 or k = –7 Solve the equation. |6x| – 8 = 22 |6x| = 30 6x = 30 or 6x = –30 x=5 or x = –5 Example 2B: Solving Absolute-Value Equations Solve the equation. Isolate the absolute-value expression. Rewrite the absolute value as a disjunction. x = 16 or x = –16 Multiply both sides of each equation by 4. Example 3A: Solving Absolute-Value Inequalities Solve the inequality. Then graph the solution. |–4q + 2| ≥ 10 –4q + 2 ≥ 10 or –4q + 2 ≤ –10 –4q ≥ 8 q ≤ –2 –3 –2 –1 0 Rewrite the absolute value as or. or –4q ≤ –12 Subtract 2 from both sides of each inequality. or q ≥ 3 Divide both sides of each inequality by –4 and reverse the inequality symbols. 1 2 3 4 5 6 Check It Out! Example 4b Solve the compound inequality. Check for Extraneous Solutions |3x +2| = 4x + 5 x = –3 or x =-1 Assignment #7 Pg 46 #12-24 x 3, 57- 66x3 HW#7 Pg 46 #12-24 x 3, 57- 66x3 Example 3B: Solving Absolute-Value Inequalities Solve the inequality. Then graph the solution. |0.5r| – 3 ≥ –3 Isolate the absolute value as or. |0.5r| ≥ 0 0.5r ≥ 0 or 0.5r ≤ 0 r≤0 Rewrite the absolute value as or. Divide both sides of each inequality by 0.5. or r ≥ 0 The solution is all real numbers, R. (–∞, ∞) –3 –2 –1 0 1 2 3 4 5 6 The solutions of |x| < 3 are the points that are less than 3 units from zero. The solution is an “and” statement: –3 < x < 3. The solutions of |x| > 3 are the points that are more than 3 units from zero. The solution is an “or” statement: x < –3 or x > 3. Note: The symbol ≤ can replace <, and the rules still apply. The symbol ≥ can replace >, and the rules still apply. Helpful Hint Think: Greator inequalities involving > or ≥ symbols are disjunctions. Think: Less thand inequalities involving < or ≤ symbols are conjunctions. Check It Out! Example 4b Solve the compound inequality. –2|x +5| > 10 Divide both sides by –2, and reverse the inequality symbol. |x + 5| < –5 x + 5 < –5 x+5>5 x < –10 or x > 0 Rewrite the absolute value as a conjunction. Subtract 5 from both sides of each inequality. Lesson Quiz: Part I Solve. Then graph the solution. 1. y – 4 ≤ –6 or 2y >8 –4 –3 –2 –1 0 {y|y ≤ –2 ≤ or y > 4} 1 2 3 4 5 2. –7x < 21 and x + 7 ≤ 6 {x|–3 < x ≤ –1} –4 –3 –2 –1 0 1 2 3 4 5 Solve each equation. 3. |2v + 5| = 9 2 or –7 4. |5b| – 7 = 13 +4 Lesson Quiz: Part II Solve. Then graph the solution. 5. |1 – 2x| > 7 {x|x < –3 or x > 4} –4 –3 –2 –1 0 1 2 3 4 5 6. |3k| + 11 > 8 R –4 –3 –2 –1 7. –2|u + 7| ≥ 16 0 ø 1 2 3 4 5 Put your answers into socrative student. Read carefully and make sure you answer the question. Do Now: What’s the ERROR? Do Now – Pick the correct answer