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Drill #4. Evaluate the following if a = -2 and b = ½. 1. ab – | a – b | Solve the following absolute value equalities : 2. |2x – 3| = 12 3. |5 – x | + 4 = 2 4. |x – 2| = 2x – 7. Drill #9. Solve the following equations: Check your solutions! 1. 2|x – 2| + 3 = 3
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Drill #4 Evaluate the following if a = -2 and b = ½. 1. ab – | a – b | Solve the following absolute value equalities: 2. |2x – 3| = 12 3. |5 – x | + 4 = 2 4.|x – 2| = 2x – 7
Drill #9 Solve the following equations: Check your solutions! 1. 2|x – 2| + 3 = 3 2. -3|2x + 4| + 2 = –1 3. |2x + 2| = 4x + 10
Drill #13 Solve the following absolute value equalities: 1. |3x – 3| = 12 2. 2|5 – x | + 4 = 2 3.-3|x – 1| + 1 = 1 4. |2x – 1| = 4x – 7
Drill #14 Solve the following absolute value equalities: 1. |3x + 8| = -x 2. 2|5 – x | – 3 = – 3 Solve the following inequalities and graph their solutions on a number line: 3.12 – 3x> 16 4.
1-4 Solving Absolute Value Equations: Review of major points • -isolate the absolute value (if its equal to a neg, no solutions) • -set up two cases (the absolute value is removed) • -solve each case. • -check each solution. (there can be 0, 1, or 2 solutions)
1-5 Solving Inequalities Objective: To solve and graph the solutions to linear inequalities.
Trichotomy Property Definition: For any two real numbers, a and b, exactly one of the following statements is true: a < b a = b a > b A number must be either less than, equal to, or greater than another number.
Addition and Subtraction Properties For Inequalities 1. If a > b, then a + c > b + c and a – c > b – c 2. If a < b, then a + c < b + c and a – c < b – c Note: The inequality sign does not change when you add or subtract a number from a side Example: x + 5 > 7
Multiplication and Division Properties for Inequalities For positive numbers: 1. If c > 0 and a < b then ac < bc and a/c < b/c 2. If c > 0 and a > b then ac > bc and a/c > b/c NOTE: The inequality stays the same For negative numbers: 3. If c < 0 and a < b then ac > bc and a/c > b/c 4. If c < 0 and a > b then ac < bc and a/c < b/c NOTE: The inequality changes Example: -2x > 6
Non-Symmetry of Inequalities If x > y then y < x • In equalities we can swap the sides of our equations: x = 10, 10 = x • With inequalities when we swap sides we have to swap signs as well: x > 10, 10 < x
Solving Inequalities • We solve inequalities the same way as equalitions, using S. G. I. R. • The inequality doesn’t change unless we multiply or divide by a negative number. Ex1: 2x + 4 > 36 Ex2: 17 – 3w > 35 Ex3:
Set Builder Notation Definition: The solution x > 5 written in set-builder notation: {x| x > 5} We say, the set x, such that x is greater than 5.
Graphing inequalities • Graph one variable inequalities on a number line. • < and > get open circles • < and > get closed circles • For > and > the graph goes to the right. (if the variable is on the left-hand side) • For < and < the graph goes to the left. (if the variable is on the left-hand side)
Special Cases Solve the following inequalities and graph their solutions on a number line. Ex1:3(1 – 2x) < -6(x – 1) Ex2:
Writing Inequalities from Verbal Expressions Define a variable, and write an inequality for each problem, then solve and graph the solution. Ex1: Twelve less than the product of three and a number is less than 21. Ex2: The quotient of three times a number and 4 is at least -16 Ex3: The difference of 5 times a number and 6 is greater than the number. Ex4: The quotient of the sum of a number and 6 is less than -2
