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Solve each polynomial inequality in Exercises $1-42$ and graph the solution set on a real number line. Express each solution set in interval notation.$$x^{2}-5 x+4>0$$

$(-\infty, 1) \cup(4, \infty)$

Algebra

Chapter 2

Polynomial and Rational Functions

Section 7

Polynomial and Rational Inequalities

Exponents and Polynomials

Rational Functions

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Lectures

01:57

Solve each polynomial ineq…

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02:08

02:19

01:52

01:25

03:21

01:36

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01:16

02:20

00:28

01:38

01:26

01:19

01:06

02:16

Okay, so we wanna graph and list the values of X that makes this inequality troop. Our first step is going to be two factor are polynomial. Since we have an X squared term, we know we're going to get something that looks like this. And our job is just to figure out what Ah, what are constant? Whole numbers that are gonna go in these two spots are gonna be so we know they have to multiply three other get four and some together to get negative. Five eso It looks like minus four minus. One will do that for us. Um, So what this tells us is that our function has two roots, one at positive, one one at positive fourth. And so if we want to find the spots where the ah function is strictly greater than zero, we know we're gonna exclude both of these points because since they are roots to our polynomial, uh, the value of our function at these spots is exactly zero. So it's not greater than so in about to open dots. So to signify that from here, we just need to check these intervals and see if they are positive or negative, which would tell us greater than or less than zero. So if we pick a number less than one say zero and we plug it in, we're gonna get negative four times, negative one, which is going to give us a positive value. If we pick a number from between one and four say to we're going to get one negative and one positive, which gives us a negative. And if we pick a number that's greater than four, we're gonna get a positive and a positive number. I want to buy those together. We're going to get positive. So our function could look something like or rather, yeah, so and this trunk, it's gonna be positive. And in this chunk in the middle, it'll be negative. And out here, it'll be positive. So what are answer then? The exes that are greater than zero? Is everything in this interval and everything in this interval? Not containing those two points. So how we would list that in, uh, interval notation is Ah, a round bracket, Um, followed by negative infinity on the point that we're going up to and then around bracket for up to positive infinity because we don't want to be including the points one or for in our final answer

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