How to Solve a Quadratic Inequality with Two Variables?

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In summary, the given inequality -1<2x^{2}-x<1 is equivalent to the two inequalities 0 < 2x2 - x + 1 AND 2x2 - x + 1 < 2. The solution set to this inequality is the set of x values on the graph of y = 2x2 - x for which -1 < y < 1. It can also be solved by solving the quadratic inequalities separately.
  • #1
skyturnred
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Homework Statement



-1<2x[itex]^{2}[/itex]-x<1

Homework Equations





The Attempt at a Solution



I can't seem to solve this.. I am in calculus but I get to this point at the end of a long question, and it seems so trivial that I didn't think it would be a good idea to post this in the calculus forum.

How do you solve this? I know (from wolframalpha) that the answer is -1/2<x<1 but I don't know how it got this.
 
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  • #2
skyturnred said:

Homework Statement



-1<2x[itex]^{2}[/itex]-x<1

Homework Equations





The Attempt at a Solution



I can't seem to solve this.. I am in calculus but I get to this point at the end of a long question, and it seems so trivial that I didn't think it would be a good idea to post this in the calculus forum.

How do you solve this? I know (from wolframalpha) that the answer is -1/2<x<1 but I don't know how it got this.
The inequality is equivalent to
0 < 2x2 - x + 1 < 2,
which is the same as these two inequalities:
0 < 2x2 - x + 1 AND 2x2 - x + 1 < 2

Do you know how to solve these quadratic inequalities?

Graphically, the solution set to your inequality is the set of x values on the graph of y = 2x2 - x for which -1 < y < 1.
 
  • #3
Mark44 said:
The inequality is equivalent to
0 < 2x2 - x + 1 < 2,
which is the same as these two inequalities:
0 < 2x2 - x + 1 AND 2x2 - x + 1 < 2

Do you know how to solve these quadratic inequalities?

Graphically, the solution set to your inequality is the set of x values on the graph of y = 2x2 - x for which -1 < y < 1.

OK thanks! Splitting it up into two inequalities really makes it much easier to think about.
 

FAQ: How to Solve a Quadratic Inequality with Two Variables?

What is the first step in solving the inequality -1<2x^2-x<1?

The first step in solving this inequality is to rewrite it in standard form, with the variable term on the left side and the constant term on the right side. This would make the inequality look like 2x^2-x-1<0.

How do you factor a quadratic expression to solve an inequality?

To factor a quadratic expression, you need to find two numbers that multiply to the constant term and add up to the coefficient of the variable term. In this case, the constant term is -1 and the coefficient of the variable term is -1. The two numbers that satisfy this condition are -1 and 1. So, we can rewrite the expression as 2x^2-1x+1x-1 and then factor it as (2x^2-1x)+(1x-1). This gives us the factored form of 2x(x-1)+1(x-1). Finally, we can factor out the common term (x-1) to get (2x+1)(x-1).

What are the critical points of a quadratic inequality?

The critical points of a quadratic inequality are the points where the parabola representing the inequality intersects the x-axis. These points are also known as the roots or solutions of the quadratic equation. In this case, the critical points are the solutions of the equation (2x+1)(x-1)=0, which are x=-1/2 and x=1.

How do you determine the sign of each interval in a quadratic inequality?

To determine the sign of each interval, we need to test a point from each interval in the original inequality. If the point satisfies the inequality, then that interval is part of the solution set and the sign is positive. If the point does not satisfy the inequality, then that interval is not part of the solution set and the sign is negative. In this case, we can choose the critical points -1/2 and 1 as the test points. Testing these points in the original inequality, we get (-1/2)^2-(-1/2)-1=1/4+1/2-1=0, which satisfies the inequality. Therefore, the interval (-1/2,1) is part of the solution and has a positive sign. Testing the point 1, we get 2(1)^2-1(1)-1=2-1-1=0, which does not satisfy the inequality. Therefore, the interval (1,+infinity) is not part of the solution and has a negative sign.

What is the final solution to the inequality -1<2x^2-x<1?

The final solution to this inequality is the union of the two intervals with positive signs: (-1/2,1) and (1,+infinity). So, the solution set can be represented as x∈(-1/2,1) U (1,+infinity).

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