Chapter 2: Equations and Inequalities 2.5: Inequalities

1. Chapter 2: Equations and Inequalities2.5: Inequalities Essential Question: What is one important difference between solving equations and solving inequalities?

2. 2.5: Inequalities • Interval Notation • [c, d] → all real numbers x such that c < x < d • (c, d) → all real numbers x such that c < x < d • [c, d) → all real numbers x such that c < x < d • (c, d] → all real numbers x such that c < x < d • Brackets “[“ & “]” represent that the endpoints of an inequality are included in the solution • Parenthesis “(“ & “)” represent that the endpoints of an inequality are not included • The above examples only cover compound inequalities. Any ideas how to represent simple inequalities?

3. 2.5: Inequalities • For lines going to the right of b • [b, ∞) → all real numbers x such that x > b • (b, ∞) → all real numbers x such that x > b • For lines going to the left of b • (-∞, b] → all real numbers x such that x < b • (-∞, b) → all real numbers x such that x < b • For the entire number line • (-∞, ∞) → represents all real numbers • Why don’t we use brackets around ∞?

4. 2.5: Inequalities • Basic Principles for Solving Inequalities • Add or subtract the same quantity on both sides of the inequality • Multiply or divide both sides of the inequality by the same positive quantity • Multiply or divide both sides of the inequality by the same negative quantity, and reverse the direction of the inequality • The third rule is the only change from solving traditional equalities

5. 2.5: Inequalities • Ex. 1 – Solving an Extended Linear Inequality • Written in interval notation, our solution is • -1 < x < 6 [-1, 6]

6. 2.5: Inequalities • Ex. 2 – Solving an Extended Linear Inequality • In interval notation, our solution is (-3, -1/5)

7. 2.5: Inequalities • Assignment • Page 124 • Problems 5-39, odd exercises • Show work • Answers in interval notation

8. Chapter 2: Equations and Inequalities2.5: InequalitiesDay 2 Essential Question: What is one important difference between solving equations and solving inequalities?

9. 2.5: Inequalities • Solving Other Inequalities • The solutions of an inequality of the form f(x) < g(x) consist of intervals on the x-axis where the graph of f is below the graph of g • To solve inequalities • Write the inequality in one of the following forms • f(x) > 0 f(x) > 0 f(x) < 0 f(x) < 0 • Determine the zeros of f, both real and extraneous, exactly if possible, approximately otherwise. • Determine the interval, or intervals, on the x-axis where the graph of f is above (in the case of > 0) or below (< 0) the x-axis, using +∞ and the zeros you found in step 2 as endpoints.

10. 2.5: Inequalities • Ex 3: Solving an Inequality • Solve x4 + 10x3 + 21x2 > 40x + 80 • Get the inequality equal to 0 • Subtract 40x and subtract 80 from both sides • x4 + 10x3 + 21x2 – 40x – 80 > 0 • We don’t know how to get exact solutions for a 4th degree polynomial, so we need to get approximate solutions by graphing • The zeros are at -1.53 and 1.89 • Because it’s >, we’re looking for where the line is above the x-axis • (-∞,-1.53) and (1.89, ∞) • Note that parenthesis are used because the original problem was > and not >

11. 2.5: Inequalities • Ex 4: Solving a Quadratic Inequality • Solve 2x2 + 3x – 4 < 0 • We can find exact solutions with quadratic equations. This one cannot be factored, so the quadratic function must be used.

12. 2.5: Inequalities • Ex 4: Continued • We know the zeros are and • They are about -2.35 and 0.85 in decimal form • There are two options used to determine the interval, a) graph or • Select a point between zeros and test • 2x2 + 3x – 4 < 0 • x < -2.35 [let’s use -3]: 2(-3)2 + 3(-3) – 4 = 5 • -2.35 < x < 0.85 [we’ll use 0]: 2(0)2 + 3(0) – 4 = -4 • x > 0.85 [we’ll use 1]: 2(1)2 + 3(1) – 4 = 1 • The first and last tests give us a value > 0, only the test in the middle satisfied the original inequality. • is our interval solution

13. 2.5: Inequalities • Ex 5: Solving an Inequality • Solve (x + 5)(x – 2)6(x – 8) < 0 • The zeros of the function are -5, 2, and 8 • Graphing • Graphing confirms that the graph is below x during-5 < x < 8 • Testing intervals • x < -5 [test -6, result: 3,670,016] • -5 < x < 2 [test 0, result: -2560] • 2 < x < 8 [test 3, result: -40] • x > 8 [test 9, result: 1,647,086] • Testing confirms the graphing conclusion • Interval solution is [-5, 8]

14. 2.5: Inequalities • Bonus example, solving fractional inequalities • Problem #58 • The equation has one real solution, 2x-1 = 0, x=½ • The equation has one extraneous solution • 5x + 3 = 0, x = -3/5 • Graphing • Graph is positive between -∞ and -3/5 and then ½ to ∞ • Testing intervals • x < -3/5 [test: -1, result: 3/2] • -3/5 < x < ½ [test: 0, result: -1/3] • x > ½ [test: 1, result: 1/8] • Both confirm an interval solution of [-∞,-3/5) and [½, ∞]

15. 2.5: Inequalities • Assignment • Page 124-125 • Problems 41-69, odd exercises