A. Determine any asymptotes, holes, and intercepts of A. asymptotes: x = 0, x = –1 and y = 1; intercept: (3, 0) B. asymptotes: x = 1 and y = 3; hole: x = –1; intercept: (3, 0); C. asymptotes: x = –1 and y = 0; hole: x = 0; intercept (0, 3); D. asymptotes: x = 0 and y = 1; hole: x = –1; intercept: (3, 0); 5–Minute Check 2

A. Determine any asymptotes, holes, and intercepts of A. asymptotes: x = 0, x = –1 and y = 1; intercept: (3, 0) B. asymptotes: x = 1 and y = 3; hole: x = –1; intercept: (3, 0); C. asymptotes: x = –1 and y = 0; hole: x = 0; intercept (0, 3); D. asymptotes: x = 0 and y = 1; hole: x = –1; intercept: (3, 0); 5–Minute Check 2

You solved polynomial and rational equations. (Lessons 2-3 and 2-4) Solve polynomial inequalities. Solve rational inequalities. Then/Now

polynomial inequality sign chart rational inequality Vocabulary

Solve a Polynomial Inequality Subtracting 1 from each side, you get x 2 – 8x + 15 ≤ 0. Let f (x) = x2 – 8x + 15. Factoring yields f (x) = (x – 5)(x – 3), so f (x) has real zeros at x = 5 and x = 3. Create a sign chart using these zeros. Then substitute an x-value in each test interval into the factored form of the polynomial to determine if f (x) is positive or negative at that point. Example 1

Think: (x – 5) and (x – 3) are Solve a Polynomial Inequality f (x) = (x – 5)(x – 3) Think: (x – 5) and (x – 3) are both negative when x = 2. f (x) = (x – 5)(x – 3) Example 1

Solve a Polynomial Inequality Because f (x) is negative in the middle interval, the solution of x 2 – 8x + 16 ≤ 1 is [3, 5]. The graph of f (x) supports this conclusion, because f (x) is below the x-axis on this same interval. Answer: [3, 5] Example 1

Solve x 2 – 9x + 10 < 46. A. (–3, 12) B. C. D. (–12, 3) Example 1

Solve x 2 – 9x + 10 < 46. A. (–3, 12) B. C. D. (–12, 3) Example 1

Solve a Polynomial Inequality Using End Behavior Solve x 3 – 22x > 3x 2 – 24. Step 1 Subtract 3x 2 – 24 from each side to get x 3 – 3x 2 – 22x + 24 > 0. Step 2 Let f (x) = x3 – 3x2 – 22x + 24. Use the techniques from Lesson 2-4 to determine that f has real zeros 1 at x = –4, x = 1, and x = 6. Set up a sign chart. Example 2

Solve a Polynomial Inequality Using End Behavior Step 3 Determine the end behavior of f (x). Because the degree of f is odd and its leading coefficient is positive, you know This means that the function starts off negative at the left and ends positive at the right. Example 2

Solve a Polynomial Inequality Using End Behavior Step 4 Because each zero listed is the location of a sign change, you can complete the sign chart. The solutions of x3 – 3x2 – 22x + 24 > 0 are the x-values such that f (x) is positive. From the sign chart, you can see that the solution set is (–4, 1)  (6, ∞). Example 2

Solve a Polynomial Inequality Using End Behavior Answer: (–4, 1)  CHECK The graph of f (x) = x 3 – 3x 2 – 22x + 24 is above the x-axis on (–4, 1)  (6, ∞). Example 2

Solve 2x3 + 9x 2 ≥ – 3x + 4. A. (–∞, –4]  B. (–∞, –4)  C. [–4, –1] or D. (–4, –1) or Example 2

Solve 2x3 + 9x 2 ≥ – 3x + 4. A. (–∞, –4]  B. (–∞, –4)  C. [–4, –1] or D. (–4, –1) or Example 2

Polynomial Inequalities with Unusual Solution Sets A. Solve x 2 + 2x + 3 < 0. The related function f (x) = x 2 + 2x + 3 has no real zeros, so there are no sign changes. This function is positive for all real values of x. Therefore, x 2 + 2x + 3 < 0 has no solution. The graph of f (x) supports this conclusion, because the graph is never below the x-axis. Answer:  Example 3

Polynomial Inequalities with Unusual Solution Sets B. Solve x 2 + 2x + 3 ≥ 0. Because the related function f (x) = x 2 + 2x + 3 is positive for all real values of x, the solution set of x 2 + 2x + 3 ≥ 0 is all real numbers or (–∞, ∞). Answer: Example 3

Polynomial Inequalities with Unusual Solution Sets C. Solve x 2 + 12x + 36 > 0. The related function f (x) = x2 + 12x + 36 has one real zero, x = –6, with multiplicity 2, so the value of f (x) does not change signs. This function is positive for all real values of x except x = –6. Therefore, the solution of x 2 + 12x + 36 > 0 is (–∞, –6)  (–6, ∞). The graph of f (x) supports this conclusion. Answer: (–∞, –6)  (–6, ∞) Example 3

Polynomial Inequalities with Unusual Solution Sets D. Solve x 2 + 12x + 36 ≤ 0. The related function f (x) = x 2 + 12x + 36 has a zero at x = –6. For all other values of x, the function is positive. Therefore, x 2 + 12x + 36 ≤ 0 has just one solution, x = –6. Answer: {–6} Example 3

Solve x 2 + 6x + 9 > 0. A. no solution B. (–∞, ∞) C. x = –3 D. (–∞, –3)  (–3, ∞) Example 3

Solve x 2 + 6x + 9 > 0. A. no solution B. (–∞, ∞) C. x = –3 D. (–∞, –3)  (–3, ∞) Example 3

Use the LCD, (x + 2) to rewrite each fraction. Then add. Solve a Rational Inequality Solve . Original inequality Use the LCD, (x + 2) to rewrite each fraction. Then add. Simplify. Example 4

Solve a Rational Inequality Let The zeros and undefined points of the inequality are the zeros of the numerator, none, and denominator, x = –2. Create a sign chart using this number. Then choose and test x-values in each interval to determine if f (x) is positive or negative. Example 4

Solve a Rational Inequality Example 4

Solve a Rational Inequality The solution set of the original inequality is the interval for which f (x) is positive, (–∞, –2). The graph of shown here supports this conclusion. Answer: Example 4

Solve . A. (–∞, 3)  [11, ∞) B. [–∞, 3]  [11, ∞) C. (3, 11] D. [3, 11] Example 4

Solve . A. (–∞, 3)  [11, ∞) B. [–∞, 3]  [11, ∞) C. (3, 11] D. [3, 11] Example 4

Solve a Rational Inequality CARPENTRY A carpenter is making tables. The tables have rectangular tops with a perimeter of 20 feet and an area of at least 24 square feet. Write and solve an inequality that can be used to determine the possible lengths of the tables. Let l represent the length of the table top. Let w represent the width of the table top. Thus, 2w + 2l = 20. Then 10 – l represents the width of the table top. Example 5

l(10 – l) ≥ 24 Write the inequality. Solve a Rational Inequality l(10 – l) ≥ 24 Write the inequality. 10l – l 2 ≥ 24 Distributive Property 0 ≥ l 2 – 10l + 24 Subtract 10l – l 2 from each side. 0 ≥ (l – 6)(l – 4) Factor. Example 5

Solve a Rational Inequality The zeroes of the related function are 6 and 4. Use these numbers to create and complete a sign chart for this function. f(x) = (l – 6)(l – 4) f(x) = (l – 6)(l – 4) Example 5

So, the solution set of l(10 – l) ≥ 24 is [4, 6]. Solve a Rational Inequality So, the solution set of l(10 – l) ≥ 24 is [4, 6]. The length must be between 4 feet and 6 feet, inclusive. Answer: l(10 – l) ≥ 24; 4 ft to 6 ft Example 5

A. l(36 – l) ≥ 80; 0 ft < l ≤ 8 ft or l ≥ 10 ft GARDENING A gardener is marking off rectangular garden plots. The perimeter of each plot is 36 feet and the area is at least 80 square feet. Write and solve an inequality that can be used to find the possible lengths of each plot. A. l(36 – l) ≥ 80; 0 ft < l ≤ 8 ft or l ≥ 10 ft B. l(18 – l) ≥ 80; 8 ft ≤ l ≤ 10 ft C. I 2 – 36l ≥ 80; 0 ft < l ≤ 8 ft or 10 ft ≤ l ≤ 36 ft D. l(18 – l) ≥ 80; 4 ft ≤ l ≤ 5 ft Example 5

A. l(36 – l) ≥ 80; 0 ft < l ≤ 8 ft or l ≥ 10 ft GARDENING A gardener is marking off rectangular garden plots. The perimeter of each plot is 36 feet and the area is at least 80 square feet. Write and solve an inequality that can be used to find the possible lengths of each plot. A. l(36 – l) ≥ 80; 0 ft < l ≤ 8 ft or l ≥ 10 ft B. l(18 – l) ≥ 80; 8 ft ≤ l ≤ 10 ft C. I 2 – 36l ≥ 80; 0 ft < l ≤ 8 ft or 10 ft ≤ l ≤ 36 ft D. l(18 – l) ≥ 80; 4 ft ≤ l ≤ 5 ft Example 5

polynomial inequality sign chart rational inequality Vocabulary