 INEQUALITIES QUADRATIC INVOLVING FUNCTIONS We are going to use what we’ve learned about the graphs of quadratic functions to solve quadratic inequalities.

You walk directly east from your house one block. How far from your house are you? 1 block You walk directly west from your house one block. How far from your house are you? It didn't matter which direction you walked, you were still 1 block from your house. This is like absolute value. It is the distance from zero. It doesn't matter whether we are in the positive direction or the negative direction, we just care about how far away we are units away from 0

What we are after here are values of x such that they are 6 away from and -6 are both 6 units away from 0 The "stuff" inside the absolute value signs could = 10 (the positive direction) or the "stuff" inside the absolute value signs could = -10 (the negative direction) Let's check it:

Absolute-Value Principle for Equations For any positive number p and any algebraic expression x: a)The solution of |X| = p are those numbers that satisfy X = -p or X = p b)The equation |X| = 0 is equivalent to the equation X = 0 c)The equation |X| = -p has no solution.

For the solution set: |2x + 5| = 13 Solution Lets use the absolute-value principle, knowing the 2x + 5 must be either 13 or -13 or The number 2x + 5 is 13 units from zero if x is replaced with -9 or 4. The solution set is {-9, 4}

For the solution set: |4 – 7x| = -8

Given that f(x) = 2|x + 3| +1, find all x for which f(x) =15 Solution Since we are looking for f(x) = 15, we substitute. The solution set is {-10, 4} Replacing f(x) with 2|x + 3| + 1 Subtracting 1 from both sides Dividing both sides by 2 Using the absolute-value principle for equations. or

Let's look at absolute value with an inequality. This is asking, "For what numbers is the distance from 0 less than 5 units?" Everything inbetween these lines is less than 5 units away from 0 Inequality notationInterval notation So if we have it is equivalent to This means x is greater than -a and x is less than a (or x is inbetween -a and a)

What if the inequality is greater than? This is asking, "When is our distance from 0 more than 5 units away?" Everything outside these lines is more than 5 units away from 0 So if we have it is equivalent to Everything outside these lines is more than 5 units away from 0 We'll have to express this with two difference pieces OR In interval notation:

This means if there are other things on the left hand side of the inequality that are outside of the absolute value signs, we must get rid of them first. We must first isolate the absolute value From what we saw previously, the "stuff" inside the absolute value is either less than or equal to -6 or greater than or equal to 6 Isolate x, remembering that if you multiply or divide by a negative you must turn the sign. - 2 We are dividing by a negative so turn the signs! or []

Consider the inequality: Let’s look at the graph It is a parabola with x intercepts (5, 0) and (-2, 0) and y intercept (0, -10). We could also find the vertex by FOILing and then using –b/2a but we will see that this will not be necessary to solve the inequality.

is still the problem we want to solve We have a graph of: Since the left hand side of the inequality is f(x), the inequality is asking, “Where is f(x) > 0”? Look at the graph to answer this question. f(x) is the y value so where is the function value or y value above the x axis? (- ,-2) or (5,  ) Since the left hand side of the inequality is f(x), the inequality is asking, “Where is f(x) > 0”?

Let’s try another one: First of all we want the right hand side to be zero so that when we look at the graph we are looking where the function is either above or below the x axis depending on the inequality. Factor Graph f(x) = left hand side, by finding intercepts Where is this graph less than or equal to 0? [-4, -3] Using interval notation: Using inequality notation:

Let’s try another one: We want the right hand side to be 0 and then factor Graph f(x) = left hand side, by finding intercepts and knowing it is a parabola opening up. Where is this graph (the y value) greater than 0? (- , -3) or (1,  ) Using interval notation: Using inequality notation: