Warm Up

Basic Inequalities Review

= / Notation < > Less than Greater than Less than or equal to Greater than or equal to = / Not equal to

Notes Graphing Inequalities Strictly greater than: a > b Strictly less than: a < b Greater than or equal to: a ≥ b Less than or equal to: a ≤ b Graphing Inequalities

Solving Inequalities inequality Solving an _________ is the same as solving an __________. equation

A new rule If you __________ or _________ a negative number you must _____ the inequality sign!!!!! multiply by divide by flip

Solving Polynomial Inequalities

Solve Equations by Factoring 1. Set the equation equal to zero. 2. Factor the polynomial completely. 3. Set each factor equal to zero and solve.

Quadratic Formula:

terminology zero solution x-intercept root Graphically Algebraically where the graph crosses the x-axis. where the function equals zero. f(x) = 0

Interval Notation A parenthesis means “do NOT include” A bracket means “INCLUDE this number” (3, 49) the numbers between 3 & 49 but not including 3 & 49

Quadratic Inequalities

Quadratics Before we get started let’s review. A quadratic equation is an equation that can be written in the form , where a, b and c are real numbers and a cannot equal zero. In this lesson we are going to discuss quadratic inequalities.

Quadratic Inequalities What do they look like? Here are some examples:

Quadratic Inequalities When solving inequalities we are trying to find all possible values of the variable which will make the inequality true. Consider the inequality We are trying to find all the values of x for which the quadratic is greater than zero or positive.

Solving a quadratic inequality algebraically We can find the values where the quadratic equals zero by solving the equation,

Solving a quadratic inequality algebraically For the quadratic inequality, we found zeros 3 and –2 by solving the equation . *To find the solution set you must test a value. ALWAYS USE “0” to determine where to shade. -2 3

Solving a quadratic inequality graphically If the inequality is: - TRUE, shade where “0” is. - FALSE, shade where “0” is not. Solution: The intervals

Example 2: Solve First find the zeros by solving the equation,

Example 2: Solution: The interval makes up the solution set for the inequality .

Examples: SOLVE THE FOLLOWING INEQUALITIES!

Solving a quadratic inequality graphically You may recall the graph of a quadratic function is a parabola and the values we just found are the zeros or x-intercepts. The graph of is

Solving a quadratic inequality graphically From the graph we can see that in the intervals around the zeros, the graph is either above the x-axis (positive or > 0) or below the x-axis (negative or <0). So we can see from the graph the interval or intervals where the inequality is positive.

Quadratic Inequalities Critical Number Critical Number Most parabolas can be broken up into 3 sections: 2 outer sections and 1 inner section. A solution set for a quadratic inequality will be either the 2 outer sections or the 1 inner section.

Quadratic Inequalities x2 + 12x + 32 < 0 -x2 – 12x – 32 < 0 -8 < x < -4 x < -8 or x > -4 (-8, -4) (- , -8) U (-4, )

Quadratic Inequalities x2 + 12x + 32 > 0 -x2 – 12x – 32 > 0 x < -8 or x > -4 -8 < x < -4 (- , -8] U [-4, ) [-8, -4]

Quadratic Inequalities -(x + 7)2 – 6 < 0 (x + 8)2 + 6 < 0 everywhere No where (- , ) These parabolas are all or nothing.

Special Cases: *Double roots *When “a” is negative…

When “a” is negative… you must divide by a negative to make “a” positive Ex. -x2 – 12x – 32 < 0

Try this on your own… -x2 – 12x – 27 > 0

Quadratic Inequalities: Double Roots Critical Number (x – 2)2 < 0 (x – 5)2 > 0 Only at one place Everywhere except 5 x = 2 x = 5 [2] (-, 5) U (5, ) These parabolas could be all or nothing.

Example 3: Solve the inequality . First find the zeros.

Example 3: But these zeros , are complex numbers. What does this mean? Let’s look at the graph of the quadratic,

Example 3: We can see from the graph of the quadratic that the curve never intersects the x-axis and the parabola is entirely below the x-axis. Thus the inequality is always true.

Summary In general, when solving quadratic inequalities Find the zeros by solving the equation you get when you replace the inequality symbol with an equals. Find the intervals around the zeros by graphing it in your calculator The solution is the interval or intervals which make the inequality true.

Tic-Tac-Know: Complete 3 in a row, your choice! Name: ________________

Practice Problems