QuestionSolve the inequality $x^2 - 2x - 3 > 0$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to solve the inequality $x^-x-3>0$ . . The inequality is quadratic, and we can solve it by finding the roots of the corresponding quadratic equation $x^-x-3=0$ .

STEP 2

First, we need to solve the quadratic equation $x^2-2x-=0$ . We can do this by factoring the quadratic equation.
$x^2-2x-=(x-a)(x-b)$

STEP 3

To find the values of $a$ and $b$ , we need to find two numbers that multiply to $-3$ (the constant term) and add to $-2$ (the coefficient of $x$ ). These numbers are $-3$ and $1$ .
So, we have $x^2-2x-3=(x-3)(x+1)$

STEP 4

Setting each factor equal to zero gives us the roots of the equation.
$x-3=0 \Rightarrow x=3$ $x+1=0 \Rightarrow x=-1$

STEP 5

Now that we have the roots of the equation, we can determine the intervals of $x$ for which the inequality $x^2-2x-3>0$ holds true. We divide the number line into three intervals using the roots $-1$ and $3$ as boundaries1. $x<-1$
2. $-1<x<3$
3. $x>3$

STEP 6

We choose a test point in each interval and substitute it into the inequality $x^2-2x-3>0$ to see if the inequality holds true.
For $x<-1$ , let's choose $x=-2$ $(-2)^2-2(-2)-3=4+4-3=5>0$ So, the inequality holds true for $x<-1$ .

STEP 7

For $-1<x<3$ , let's choose $x=0$ $0^2-2(0)-3=-3<0$ So, the inequality does not hold true for $-1<x<3$ .

STEP 8

For $x>3$ , let's choose $x=4$ $(4)^2-2(4)-3=16-8-3=5>0$ So, the inequality holds true for $x>3$ .

STEP 9

From steps6,7, and8, we conclude that the solution to the inequality $x^2-2x-3>$ is $x<-$ or $x>3$ .
The solution to the inequality $x^2-2x-3>$ is $x<-$ or $x>3$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms