Solving Inequalities > < < > > < Set Builder Notation { x| x < –2.7; x  R } created & designed by Tammy the Tutor © MathRoom Learning Service 2011.

____________________ Instructions INSTRUCTIONS Solve for Real Number values of x : Show all your work. a) Write the solution as an interval. b) Draw a number line of the solution. Click your mouse to see the solution. ____________________

Question 1 step 1: transpose terms to get – 4x < – 12 1. step 1: transpose terms to get – 4x < – 12 step 2: divide both sides by – 4 x > 3 0 1 2 3 4 5 6 7 solution: ] 3, ∞ [

Question 2 2. step 1: transpose terms to get – 5x < 4 step 2: divide both sides by – 5 x > – 4/5 solution: ] – 4/5, ∞ [ –2 –1 0 1 2 3 4

Question 3 step 1: multiply through by 12: – 10x < 9 + 32x 3. step 1: multiply through by 12: – 10x < 9 + 32x step 2: collect like terms – 42x < 9 step 3: divide by – 42 x > – 9/42 or x > – 3/14 –1 0 1 2 3 solution: [–3/14, ∞[

Question 4 step 1: remove brackets: 4x² – 8x < 4x² – 14x + 6 4. step 1: remove brackets: 4x² – 8x < 4x² – 14x + 6 step 2: collect like terms to get: 6x < 6 step 3: divide both sides by 6 x < 1 solution: ] – ∞, 1 [ –2 –1 0 1 2

Question 5 step 1: transpose, collect terms: 12x > – 5 5. step 1: transpose, collect terms: 12x > – 5 step 2: divide both sides by 12 x > – 5/12 solution: [ –5/12, ∞ [ –2 –1 0 1 2 3 4

Set Builder Notation 1-2 { x| x > 2.5; x  R } Write Set Builder Notation for these: All sets are from the Real Numbers (R) Click your mouse to see the solution. 1. { x| x > 2.5; x  R } 2. { y| y < –7; y  R }

Set Builder Notation 3-4 step 1: transpose terms to get m³ – m² = – 5 3. step 1: transpose terms to get m³ – m² = – 5 solution { m| m³ – m² = – 5; m  R } 4. step 1: divide by 2 to get t² = 5 solution { t| t² = 5; t  R }

____________________ Word Problem #1 Find all Natural Numbers such that 5 more than 3 times the number is greater than 4, but less than or equal to 27. ____________________

Problem #1: Set-Up We need Natural Numbers such that 5 more than 3 times the number is greater than 4, but less than or equal to 27. ____________________ step 1: name the variable (make a “ let ” statement) let n = the number step 2: translate words to a math statement: 4 < 3n + 5  27

Problem #1: Solution step 3: solve the inequation: 4 < 3n + 5  27 since n is a Natural Number, 0 < n  7 The numbers are 1, 2, 3, 4, 5, 6, or 7

Word Problem #2 x – 4 x – 2 The area of this rug is greater than 3 m² but less than or equal to 15 m². a) Find integer values for x. b) Find the exact area. ____________________

Problem #2: Set-Up x – 4 x – 2 a) Find integer values for x if the Area > 3 but  15. Area = (x – 4)(x – 2) SOLUTION: step 1: write the inequality: 3 < (x – 4)(x – 2)  15 step 2: simplify the inequality: 3 < x² – 6x + 8  15

Problem #2: Solution step 3: solve 2 separate quadratic inequalities 0 < x² – 6x + 5 x² – 6x – 7  0 ( x – 5)(x – 1) > 0 ( x – 7)(x + 1)  0 x < 1, or x > 5 x > –1, or x  7 Find where these solutions intersect with a number line. –2 –1 0 1 2 3 4 5 6 7 x – 4 x – 2 a) integer values for x are 6 and 7 b) the exact area is 8m² if x = 6 and 15m² if x = 7.