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23 Apr 2014, 11:57
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Solving Quadratic Inequalities: Graphical Approach
Suppose we need to find the range of values for \(x\) when \(x^2-4x+3<0\). The equation \(x^2-4x+3=0\) represents the graph of a parabola, which looks like this:
The intersection points are the roots of the equation \(x^2-4x+3=0\), which are \(x_1=1\) and \(x_2=3\). The "<" sign indicates the range of \(x\) where the graph is below the x-axis. The answer is \(1<x<3\) (between the roots).
If the sign were ">": \(x^2-4x+3>0\). First, find the roots (\(x_1=1\) and \(x_2=3\)). The ">" sign indicates the range of \(x\) where the graph is above the x-axis. The answer is \(x<1\) and \(x>3\) (to the left of the smaller root and to the right of the larger root).
This approach works for any quadratic inequality. For example, consider \(-x^2-x+12>0\). First, rewrite this as \(x^2+x-12<0\) (so that the coefficient of x^2 is positive. It's possible to solve without rewriting, but it's easier to master one specific pattern).
For \(x^2+x-12<0\), the roots are \(x_1=-4\) and \(x_2=3\). The "<" sign indicates that the range for \(x\) is below the x-axis, resulting in \(-4<x<3\) (between the roots).
On the other hand, if the inequality were \(x^2+x-12>0\), the answer would be \(x<-4\) and \(x>3\) (to the left of the smaller root and to the right of the larger root).
Show SpoilerCheck more on inequalities under the spoiler
9. Inequalities
- Theory
Inequalities Made Easy!
Inequalities: Tips and hints
Arithmetic with Inequalities
Wavy Line Method Application - Complex Algebraic Inequalities
Solving Quadratic Inequalities: Graphic Approach
Inequalities - Quadratic Inequalities
Graphic approach to problems with inequalities
Inequalities trick
INEQUATIONS(Inequalities) Part 1
INEQUATIONS(Inequalities) Part 2
28 May 2014, 04:23
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Check out all important topics we have in Main Math forum HERE.
Important topics in PS forums.
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Important topics in PS forums.
Important topics in DS forums.
General Discussion
pearljiandani
Joined: 21 Oct 2019
Last visit: 03 Jul 2024
Posts: 23
Given Kudos: 22
Location: India
Schools: Kellogg '25 (M$) Ross '25 (A$)
GMAT 1: 720 Q49 V39
GPA: 3.64
04 Dec 2020, 03:18
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Hi Bunuel
I read this post of yours and it was extremely helpful. However I'm a little confused, it'll be great if you can help me out here. So in this post, it says that if the sign is "<", it means that x lies between the 2 roots and if the sign is ">", then x lies on the left of the smaller root and on the right of the bigger root. Now I was solving this other question -
If x^2 + 2x -15 = -m, where x is an integer from -10 and 10, inclusive, what is the probability that m is greater than zero?
A. 2/7
B. 1/3
C. 7/20
D. 2/5
E. 3/7
Re-arrange the given equation: −x^2−2x+15=m
Given that xx is an integer from -10 and 10, inclusive (21 values) we need to find the probability that −x^2−2x+15 is greater than zero, so the probability that −x^2−2x+15>0
Factorize: (x+5)(3−x)>0. This equation holds true for −5<x<3
Since x is an integer then it can take the following 7 values: -4, -3, -2, -1, 0, 1, and 2.
Here, if (x+5)(3−x)>0, then shouldn't x be smaller than -5 and greater than 3? I know that doesn't satisfy the equation given in the question but I was wondering how this was possible if the sign ">" meant that the solution lies on the left of the smaller root and on the right of the bigger root.
I read this post of yours and it was extremely helpful. However I'm a little confused, it'll be great if you can help me out here. So in this post, it says that if the sign is "<", it means that x lies between the 2 roots and if the sign is ">", then x lies on the left of the smaller root and on the right of the bigger root. Now I was solving this other question -
If x^2 + 2x -15 = -m, where x is an integer from -10 and 10, inclusive, what is the probability that m is greater than zero?
A. 2/7
B. 1/3
C. 7/20
D. 2/5
E. 3/7
Re-arrange the given equation: −x^2−2x+15=m
Given that xx is an integer from -10 and 10, inclusive (21 values) we need to find the probability that −x^2−2x+15 is greater than zero, so the probability that −x^2−2x+15>0
Factorize: (x+5)(3−x)>0. This equation holds true for −5<x<3
Since x is an integer then it can take the following 7 values: -4, -3, -2, -1, 0, 1, and 2.
Here, if (x+5)(3−x)>0, then shouldn't x be smaller than -5 and greater than 3? I know that doesn't satisfy the equation given in the question but I was wondering how this was possible if the sign ">" meant that the solution lies on the left of the smaller root and on the right of the bigger root.
04 Dec 2020, 04:50
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pearljiandani
Rewrite (x+5)(3−x)>0 as (x+5)(x−3)<0 (so that you have (x +/- something )(x +/- something) ) --> -5 < x < 3.
Or if you expand: (x+5)(3−x)>0 you get -x^2 - 2 x + 15 > 0. Rewrite as x^2 + 2x - 15 < 0. --> roots are -5 and 3 and < indicates between the roots: -5 < x < 3.
Hope it's clear.
