inequality

[triclino]

For what values of a and b does the following inequality hold:

 

[math]a^2+ab-4a+3>0[/math]

[shyvera]

If the determinant is negative (treating the LHS as a quadratic in a), then the inequality holds for all real values of a. Otherwise it holds for all a such that [math]a<\alpha_1[/math] or [math]a>\alpha_2[/math] where [math]\alpha_1,\ \alpha_2[/math] are the real roots of the equation [math]x^2+(b-4)x+3=0[/math] with [math]\alpha_1\leqslant\alpha_2.[/math]

[triclino]

If the determinant is negative (treating the LHS as a quadratic in a), then the inequality holds for all real values of a.

 

How about, b .

 

For what values of b does the inequality hold??

[shyvera]

That’s what you’re supposed to calculate! I didn’t post the full solution lest the moderators are not happy with me for doing your homework for you (and being a new member I’m trying my best not to make a bad impression).

 

You should find that the determinant is negative if b is within a certain range of values. I make it [hide]4−2√3< b < 4+2√3[/hide] In this case, the inequality is satisfied for all real values of a. If b is outside that range of values, solve the quadratic equation for the real roots, expressed in terms of b. Then the inequality is satisfied when a is less than the lesser of the two roots or greater than the greater of the two roots.

Edited May 27, 2010 by shyvera

[mooeypoo]

Moved to homework help.

 

triclino, you are not new nin this forum, you know the rules and you know how we operate. If you want help, show what you already did and where you got stuck and we'd love to help you. We're not here to solve the question for you.

 

~moo

[triclino]

You should find that the determinant is negative if b is within a certain range of values. I make it [hide]4−2√3< b < 4+2√3[/hide] In this case, the inequality is satisfied for all real values of a.

 

 

How about a=0 .Is not the inequality satisfied for all values of b??

[D H]

How about a=0 .Is not the inequality satisfied for all values of b??

shyvera chose to look at the problem from the perspective of finding the values of a that make the inequality true given some particular value of b. You can certainly look at it from the alternative perspective, finding the values of b that make the inequality true given some particular value of a.

 

Both points of view yield the same set, just described in two different ways.