Section 4.1 Inequalities & Applications. Inequalities. Equations. 3x + 7 = 13 y = 7 3x + 2y = 6. 3x + 7 < 13 y > 7 3x + 2y 6. Symbols: < > 4.

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1 Section 4.1 Inequalities & Applications Equations 3x + 7 = 13 y = 7 3x + 2y = 6 Inequalities 3x + 7 < 13 y > 7 3x + 2y 6 Symbols: < > 4.1 1

2 Overview of Linear Inequalities 4.1 Study Inequalities with One Variable Why study Inequalities? How do we show a Solution to an Inequality With an Algebraic Statement -5 < x x 11 x -1 < 3 By Graphing Using the Number Line Using Interval Notation (-5, ) (-, 11] (-,-2)U(2, ) Via Set-Builder Notation { x x > -5} {x -4 <x 13 } Use Properties to Solve Single-Variable Inequalities ( 4.1 2

3 Solving Inequalities Examples of Inequalities containing a variable -2 < a x > 4 x y 10y 4 6 < x + 2 < 9 A solution is a value that makes an inequality true A solution set is the set of all solutions Substitute a value to see if it is a solution: Is 5 a solution to x + 3 < 6 no, 8 < 6 is false Is -1 a solution to -3 > -5 2x no, -3 > -3 false Is 2 a solution to y yes, 4 4 is true 3

4 Showing an Inequality Solution on the Number Line Identify the Boundary Point on the number line Left ( or Right ) : the number is not a solution Left [ or Right ] : the number is a solution Shade the number line over all solutions ) 4.1 4

5 Interval Notation Is a compact way to precisely specify a complete solution set over a range of numbers, without reference to a variable name Parentheses ( ) and Brackets [ ] have another use They are used to enclose a pair of values The smallest number is always listed on the left They have the same meaning as used in graphing: point included or excluded Examples: (2, 5) [0, 15) (-4, -2] [-22, 43] (-, 7) and [3, ) show how infinity values are noted Interval notation can be used in addition to Set-builder 4.1 5

6 Example: Number Line Graphs and Notation What is the inequality for all real numbers greater than -5? Either of the algebraic statements x > -5 or -5 < x Set-builder Notation (using the inequality) can also be used { x -5 < x } We can also show this interval in Interval Notation as (-5, ) We can graph this by identifying a portion of the number line. Any portion of a number line is called an interval For example, all real numbers greater than -5 up to is graphed ( 4.1 6

7 ( or ) [ or ] Unbounded Intervals ( point not included [ point is included ( a [ a ) a ] a a Some textbooks use a hollow dot in place of ( or ) and a solid dot in place of [ or ] 4.1 7

8 Principles Used to Solve Inequalities Addition (and Subtraction) Principles Any real number (or term) can be added to (or subtracted from) both sides of an inequality to produce another inequality with the same solutions. The direction of the inequality symbol is unchanged. a < b is equivalent to a + c < b + c (true for >,, < and ) Example 4 Solve and graph a) x + 5 > 1 x > -4 (-4, ) {x x>-4} b) 4x 1 5x 2 1 x (-,1] {x x 1} 4.1 8

9 Principles Used to Solve Inequalities Positive Multiplication (and Division) Principles If both sides of an inequality are multiplied (or divided) by a positive number, the new inequality will have the same solutions. The direction of the inequality symbol is unchanged. a < b is equivalent to a c < b c (true for >,, < and ) Example 5 Solve and graph a) 3y < ¾ y < ¼ (-,¼) {y y<¼} b) ½t 5 t 10 [10, ) {t t 10} 4.1 9

10 Principles Used to Solve Inequalities Negative Multiplication (and Division) Principles If both sides of an inequality are multiplied or divided by a negative number, the new inequality will be equivalent only if the direction of the inequality symbol is reversed! Consider 5 < 7 Let s multiply by -1: (-1)5 < (-1)7 is -5 < -7? No! -5 > -7 so remember to reverse the symbol Example 5 Solve and graph a) -5y < 3.2 y > (-0.64, ) {y y>-0.64} b) - ¼x -80 x 320 (-,320] {x x 320}

11 Principles Used Together As in equations, First use the addition principle to move variable terms to one side and numeric terms to the other Then use the multiplication principle to effect the solution Example 6 Solve and graph a) y 10y 2 y 6 (-,6] {y y 6 } b) -3(x + 8) 5x > 4x 12 x < 1 (-,1) {x x < 1 }

12 Class Practice Use the properties of linear inequalities to solve the following and graph the solution set. Also give the solution in interval notation. 2y y 8 y 4 [4, ) [ -9(h 3) + 2h < 8(4 h) x 3 x 3 x 1 9h ( 7h 8h x 3) x 27 7x 9 h 2h 32 8h 927 x 9 8h 7x 9 x 5 ( x 16x 3) x h 5( x 3) 12( x 3) 9x 9 5x 15 12x 36 9x

13 Problem Solving The statements a does not exceed b and a is at most b both translate to the inequality a b The statements a is at least b and a is no less than b both translate to the inequality a b

14 Applications Truck Rentals. Campus Fun rents a truck for $45 plus 20 per mile. Their rental budget is $75. For what mileages will they NOT exceed their budget? for 150 miles or less or mileage less than or equal to 150 Charges = f(m) = m m

15 What Next? Section 4.2 Intersections, Unions & Compound Inequalities