Definition of a Polynomial Inequality Date: Topic: Solving Inequalities in One Variable (2.9) Definition of a Polynomial Inequality A polynomial inequality is any inequality that can be put in one of the forms f(x) < 0, f(x) > 0, f(x) ≤ 0, or f(x) ≥ 0 where f is a polynomial function.

You can also view this on a graphing calculator. Determine the x values that cause the polynomial function f(x) = (x + 3)(x2 + 1)(x - 4)2 to be zero, positive, and negative. real zeros? -3 and 4 Evaluate f at one number within each interval create a sign chart: (-)(+)(-)2 (+)(+)(-)2 (+)(+)(+)2 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 x negative positive positive positive? negative? You can also view this on a graphing calculator.

Solve the function for the four inequalities: Determine the x values that cause the polynomial function f(x) = (x + 3)(x2 + 1)(x - 4)2 to be zero, positive, and negative. (-)(+)(-)2 (+)(+)(-)2 (+)(+)(+)2 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 x negative positive positive Solve the function for the four inequalities: (x + 3)(x2 + 1)(x - 4)2 > 0 (x + 3)(x2 + 1)(x - 4)2 > 0 (x + 3)(x2 + 1)(x - 4)2 < 0 (x + 3)(x2 + 1)(x - 4)2 < 0

Procedure for Solving Polynomial Inequalities Express the inequality in the standard form f(x) < 0 or f(x) > 0 Find the zeros of f, the boundary points. Locate these boundary points or zeros on a number line (or sign chart), thereby dividing the number line into intervals. Evaluate f at one representative number within each interval. Write the solution set; selecting the interval(s) that satisfy the given inequality.

Solve the inequality and support it graphically: 2x2 – 3x > 2. Example Solve the inequality and support it graphically: 2x2 – 3x > 2. Step 1 Write the inequality in standard form. We can write by subtracting 2 from both sides to get zero on the right. 2x2 – 3x – 2 > 2 – 2 2x2 – 3x – 2 > 0 Step 2 Solve the related quadratic equation to find the zeros (boundary points). 2x2 – 3x – 2 = 0 This is the related quadratic equation. (2x + 1)(x – 2) = 0 Factor. 2x + 1 = 0 or x – 2 = 0 Set each factor equal to 0. x = -1/2 or x = 2 Solve for x. The boundary points or zeros are –1/2 and 2.

(-∞, -1/2], [-1/2, 2], [2, ∞) Example cont. Solve the inequality and support it graphically: 2x2 – 3x > 2. Step 3 Locate the boundary points or zeros on a number line (or sign chart). The number line with the boundary points is shown as follows: -1/2 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 x The boundary points or zeros divide the number line into three test intervals. Including the boundary points (because of the given greater than or equal to sign), the intervals are (-∞, -1/2], [-1/2, 2], [2, ∞)

Solve the inequality and support it graphically: 2x2 – 3x > 2. Step 4 Take one representative number within each test interval and substitute that number into the inequality. -1/2 2 (-)(-) (+)(-) (+)(+) -5 -4 -3 -2 -1 0 1 2 3 4 5 6 x (-∞, -1/2], [-1/2, 2], [2, ∞) True or Positive False or Negative True or Positive (-∞, -1/2] belongs to the solution set. [-1/2, 2] does not belong to the solution set. [2, ∞) belongs to the solution set.

Example cont. (-∞, -1/2] or [2, ∞) Solve the inequality and support it graphically: 2x2 – 3x > 2. Step 5 Write the solution set; selecting the interval(s) that satisfy the given inequality: (-∞, -1/2] or [2, ∞) Support it graphically.

Text Example This is the given inequality. Solve and graph the solution set: Step 1 Express the inequality so that one side is zero and the other side is a single quotient. We subtract 2 from both sides to obtain zero on the right. This is the given inequality. Subtract 2 from both sides, obtaining 0 on the right. The least common denominator is x=3. Express 2 in terms of this denominator. Subtract rational expressions. Apply the distributive property. Simplify.

Text Example cont. (-∞, -5], [-5, -3), (-3, ∞) Solve and graph the solution set: Step 2 Find boundary points by setting the numerator and the denominator equal to zero. Set the numerator and denominator equal to 0. These are the values that make the previous quotient zero or undefined. = x + 3 -x - 5 Solve for x. -3 = x x = -5 The boundary points are -5 and -3. Because equality is included in the given less-than-or-equal-to symbol, we include the value of x that causes the quotient to be zero. Thus, -5 is included in the solution set. By contrast, we do not include -3 in the solution set because -3 makes the denominator zero or undefined. Step 3 Locate boundary points on a number line (or sign chart). -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 x The boundary points divide the number line into three test intervals, namely (-∞, -5], [-5, -3), (-3, ∞)

Text Example cont. (-∞, -5], [-5, -3), (-3, ∞) false or negative Solve and graph the solution set: x -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 Step 4 Take one representative number within each test interval and substitute that number into the inequality. (-∞, -5], [-5, -3), (-3, ∞) false or negative true or positive False or negative (-∞, -5] does not belong to the solution set. [-5,-3) belongs to the solution set. (-3, ∞) does not belong to the solution set. Step 5 The solution set are the intervals that produced a true statement. Our analysis shows that the solution set is [-5, -3).

Complete Student Checkpoint Solve and support graphically. (-)(-)(-) (+)(-)(-) (+)(+)(-) (+)(+)(+) -5 -4 -3 -2 -1 0 1 2 3 4 5 x negative positive negative positive

Complete Student Checkpoint Solve and support graphically: (-)/(-) (-)/(+) (+)/(+) -5 -4 -3 -2 -1 0 1 2 3 4 5 x positive negative positive

Solving Inequalities in One Variable