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Quadratic inequalities

Inequalities

Japanese school year: Math I

What you learn

Learn how to solve quadratic inequalities by analyzing the relationship between a quadratic graph and the x-axis. This method is essential for determining ranges and domains in various mathematical applications. Reviewing factoring techniques and how to sketch parabolas beforehand will help you understand the concepts more deeply.

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Key points

This formula gives the range of xx where the quadratic expression is negative. Since the upward-opening parabola lies below the xx-axis between its roots, the solution is the open interval α<x<β\alpha < x < \beta.

(x−α)(x−β)<0  ⟺  α<x<β(α<β)(x - \alpha)(x - \beta) < 0 \iff \alpha < x < \beta \quad (\alpha < \beta)

This formula gives the range of xx where the quadratic expression is positive. Since the upward-opening parabola lies above the xx-axis outside its roots, the solution consists of the outer intervals x<αx < \alpha and β<x\beta < x.

(x−α)(x−β)>0  ⟺  x<α,  β<x(α<β)(x - \alpha)(x - \beta) > 0 \iff x < \alpha, \; \beta < x \quad (\alpha < \beta)

Because the square of any real number is always greater than or equal to 0, it can be less than or equal to 0 only when it equals 0, giving the single solution x=αx = \alpha.

(x−α)2≤0  ⟺  x=α(x - \alpha)^2 \le 0 \iff x = \alpha

Since the square of any non-zero real number is always positive, the solution to this inequality consists of all real numbers except x=αx = \alpha.

(x−α)2>0  ⟺  x≠α(x - \alpha)^2 > 0 \iff x \neq \alpha

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