What Are Rational Inequalities?

Rational inequalities are inequalities with $x$ in the denominator.

Rule

This Is How You Solve Rational Inequalities

1.: Move all of the terms over to the left-hand side.
2.: Write the expression on the left-hand side as a single fraction.
3.: Factorize the numerator and the denominator individually.
4.: Make a sign chart and find the solution.

Example 1

Solve the inequality $\frac{3 x + 7}{x + 1} > 2 x + 3$

\begin{array}{l} \frac{3 x + 7}{x + 1} & > 2 x + 3 \\ \frac{3 x + 7}{x + 1} - (2 x + 3) & > 0 \\ \frac{3 x + 7}{x + 1} - \frac{(2 x + 3) (x + 1)}{x + 1} & > 0 \\ \frac{3 x + 7 - 2 x^{2} - 2 x - 3 x - 3}{x + 1} & > 0 \\ \frac{- 2 x^{2} - 2 x + 4}{x + 1} & > 0 \\ \frac{- 2 (x^{2} + x - 2)}{x + 1} & > 0 \\ \frac{- 2 (x + 2) (x - 1)}{x + 1} & > 0 \end{array}

Now you can make a sign chart for each of the factors and combine them:

The sign chart of -2(x+2)(x-1)/(x+1) as a combination of the sign charts of its factors.

From the sign charts you can see that the expression is positive on the interval

(- \infty, - 2) \cup (- 1, 1) .

There’s a discontinuity where you have a cross in the charts. The expression isn’t defined for the value where the cross is, which means you will always have an open interval for these points. The reason for this is that you can’t use undefined points in your solution.

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How to Solve Inequalities of Degree 3 or More

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