(x - 3)/(x - 1) ≥ 0

(-infinity, 1) or [3, infinity)

Rational inequalities worksheet for Algebra 2

In Algebra 2 rational inequalities often start with a denominator that has to be factored first. Build one sign chart from the zeros of the top and the bottom together, then write the answer in interval notation, with a parenthesis at every value that makes the bottom zero.

Also taught in Precalculus, at its own level and with its own examples.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Rational Inequalities Date Period Solve each inequality. Write each answer in interval notation. $\frac{x - 3}{x^{2} - x - 20} < 0$ $\frac{x + 6}{x^{2} - 2 x - 15} \geq 0$ $\frac{x - 1}{x^{2} + 9 x + 14} \leq 0$ $\frac{x + 4}{x^{2} - 1} \leq 0$ $\frac{x + 7}{x^{2} - 3 x + 2} \leq 0$ $\frac{x + 7}{x^{2} - x - 30} \geq 0$ $\frac{x - 6}{x^{2} - 10 x + 21} > 0$ $\frac{x - 5}{x^{2} + x - 6} > 0$ $\frac{x - 2}{x^{2} - 9} < 0$ $\frac{x + 1}{x^{2} - 7 x + 6} > 0$ $\frac{x - 1}{x^{2} + x - 30} \leq 0$ $\frac{x + 5}{x^{2} - 9 x + 18} < 0$ $\frac{x + 6}{x^{2} + 10 x + 21} \geq 0$ $\frac{x + 6}{x^{2} + 4 x - 21} \leq 0$ $\frac{x + 3}{x^{2} + 8 x + 7} < 0$ $\frac{x + 2}{x^{2} - 25} < 0$ $\frac{x + 5}{x^{2} - 2 x - 24} > 0$ $\frac{x - 4}{x^{2} - 4 x - 21} \leq 0$ $\frac{x + 1}{x^{2} + x - 30} < 0$ $\frac{x + 4}{x^{2} + 3 x - 10} < 0$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Rational Inequalities

Date Period

Solve each inequality. Write each answer in interval notation.

$\frac{x - 3}{x^{2} - x - 20} < 0$
$\frac{x + 6}{x^{2} - 2 x - 15} \geq 0$
$\frac{x - 1}{x^{2} + 9 x + 14} \leq 0$
$\frac{x + 4}{x^{2} - 1} \leq 0$
$\frac{x + 7}{x^{2} - 3 x + 2} \leq 0$
$\frac{x + 7}{x^{2} - x - 30} \geq 0$
$\frac{x - 6}{x^{2} - 10 x + 21} > 0$
$\frac{x - 5}{x^{2} + x - 6} > 0$
$\frac{x - 2}{x^{2} - 9} < 0$
$\frac{x + 1}{x^{2} - 7 x + 6} > 0$
$\frac{x - 1}{x^{2} + x - 30} \leq 0$
$\frac{x + 5}{x^{2} - 9 x + 18} < 0$
$\frac{x + 6}{x^{2} + 10 x + 21} \geq 0$
$\frac{x + 6}{x^{2} + 4 x - 21} \leq 0$
$\frac{x + 3}{x^{2} + 8 x + 7} < 0$
$\frac{x + 2}{x^{2} - 25} < 0$
$\frac{x + 5}{x^{2} - 2 x - 24} > 0$
$\frac{x - 4}{x^{2} - 4 x - 21} \leq 0$
$\frac{x + 1}{x^{2} + x - 30} < 0$
$\frac{x + 4}{x^{2} + 3 x - 10} < 0$



Sigma Prep · sigmaprep.io/worksheets Name Rational InequalitiesAnswer keyVersion 2 Date Period Solve each inequality. Write each answer in interval notation. $\frac{x - 3}{x^{2} - x - 20} < 0$ $(- \infty, - 4) \cup (3, 5)$ $\frac{x + 6}{x^{2} - 2 x - 15} \geq 0$ $[- 6, - 3) \cup (5, \infty)$ $\frac{x - 1}{x^{2} + 9 x + 14} \leq 0$ $(- \infty, - 7) \cup (- 2, 1]$ $\frac{x + 4}{x^{2} - 1} \leq 0$ $(- \infty, - 4] \cup (- 1, 1)$ $\frac{x + 7}{x^{2} - 3 x + 2} \leq 0$ $(- \infty, - 7] \cup (1, 2)$ $\frac{x + 7}{x^{2} - x - 30} \geq 0$ $[- 7, - 5) \cup (6, \infty)$ $\frac{x - 6}{x^{2} - 10 x + 21} > 0$ $(3, 6) \cup (7, \infty)$ $\frac{x - 5}{x^{2} + x - 6} > 0$ $(- 3, 2) \cup (5, \infty)$ $\frac{x - 2}{x^{2} - 9} < 0$ $(- \infty, - 3) \cup (2, 3)$ $\frac{x + 1}{x^{2} - 7 x + 6} > 0$ $(- 1, 1) \cup (6, \infty)$ $\frac{x - 1}{x^{2} + x - 30} \leq 0$ $(- \infty, - 6) \cup [1, 5)$ $\frac{x + 5}{x^{2} - 9 x + 18} < 0$ $(- \infty, - 5) \cup (3, 6)$ $\frac{x + 6}{x^{2} + 10 x + 21} \geq 0$ $(- 7, - 6] \cup (- 3, \infty)$ $\frac{x + 6}{x^{2} + 4 x - 21} \leq 0$ $(- \infty, - 7) \cup [- 6, 3)$ $\frac{x + 3}{x^{2} + 8 x + 7} < 0$ $(- \infty, - 7) \cup (- 3, - 1)$ $\frac{x + 2}{x^{2} - 25} < 0$ $(- \infty, - 5) \cup (- 2, 5)$ $\frac{x + 5}{x^{2} - 2 x - 24} > 0$ $(- 5, - 4) \cup (6, \infty)$ $\frac{x - 4}{x^{2} - 4 x - 21} \leq 0$ $(- \infty, - 3) \cup [4, 7)$ $\frac{x + 1}{x^{2} + x - 30} < 0$ $(- \infty, - 6) \cup (- 1, 5)$ $\frac{x + 4}{x^{2} + 3 x - 10} < 0$ $(- \infty, - 5) \cup (- 4, 2)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Rational InequalitiesAnswer keyVersion 2

Date Period

Solve each inequality. Write each answer in interval notation.

$\frac{x - 3}{x^{2} - x - 20} < 0$ $(- \infty, - 4) \cup (3, 5)$
$\frac{x + 6}{x^{2} - 2 x - 15} \geq 0$ $[- 6, - 3) \cup (5, \infty)$
$\frac{x - 1}{x^{2} + 9 x + 14} \leq 0$ $(- \infty, - 7) \cup (- 2, 1]$
$\frac{x + 4}{x^{2} - 1} \leq 0$ $(- \infty, - 4] \cup (- 1, 1)$
$\frac{x + 7}{x^{2} - 3 x + 2} \leq 0$ $(- \infty, - 7] \cup (1, 2)$
$\frac{x + 7}{x^{2} - x - 30} \geq 0$ $[- 7, - 5) \cup (6, \infty)$
$\frac{x - 6}{x^{2} - 10 x + 21} > 0$ $(3, 6) \cup (7, \infty)$
$\frac{x - 5}{x^{2} + x - 6} > 0$ $(- 3, 2) \cup (5, \infty)$
$\frac{x - 2}{x^{2} - 9} < 0$ $(- \infty, - 3) \cup (2, 3)$
$\frac{x + 1}{x^{2} - 7 x + 6} > 0$ $(- 1, 1) \cup (6, \infty)$
$\frac{x - 1}{x^{2} + x - 30} \leq 0$ $(- \infty, - 6) \cup [1, 5)$
$\frac{x + 5}{x^{2} - 9 x + 18} < 0$ $(- \infty, - 5) \cup (3, 6)$
$\frac{x + 6}{x^{2} + 10 x + 21} \geq 0$ $(- 7, - 6] \cup (- 3, \infty)$
$\frac{x + 6}{x^{2} + 4 x - 21} \leq 0$ $(- \infty, - 7) \cup [- 6, 3)$
$\frac{x + 3}{x^{2} + 8 x + 7} < 0$ $(- \infty, - 7) \cup (- 3, - 1)$
$\frac{x + 2}{x^{2} - 25} < 0$ $(- \infty, - 5) \cup (- 2, 5)$
$\frac{x + 5}{x^{2} - 2 x - 24} > 0$ $(- 5, - 4) \cup (6, \infty)$
$\frac{x - 4}{x^{2} - 4 x - 21} \leq 0$ $(- \infty, - 3) \cup [4, 7)$
$\frac{x + 1}{x^{2} + x - 30} < 0$ $(- \infty, - 6) \cup (- 1, 5)$
$\frac{x + 4}{x^{2} + 3 x - 10} < 0$ $(- \infty, - 5) \cup (- 4, 2)$

How to do these

$\frac{x + 3}{x^{2} - 4} \geq 0$

Factor the bottom: (x - 2)(x + 2). The chart splits at -3, -2 and 2.
Test points give the signs -, +, -, + from left to right.
Include -3, where the top is zero, but not -2 or 2.

The answer is $[- 3, - 2) \cup (2, \infty)$ .

Where students go wrong

Closing the bracket at a zero of the bottom. With ≥ or ≤ the zeros of the top are included, but a value that makes the bottom zero, like -2 or 2 here, is never in the answer.

Also called solving rational inequalities, inequalities with fractions or sign chart.

What you can put on this worksheet

Rational inequalities: $\frac{x - 3}{x - 1} \geq 0$ → $(- \infty, 1) \cup [3, \infty)$
One fraction against a number: $\frac{x - 11}{x - 3} \geq 2$ → $[- 5, 3)$
A quadratic over a linear factor: $\frac{x (x - 1)}{x - 3} > 0$ → $(0, 1) \cup (3, \infty)$
A fraction on each side: $- \frac{3}{x} \geq - \frac{2}{x - 1}$ → $(- \infty, 0) \cup (1, 3]$
A squared factor: $\frac{x + 5}{x^{2}} \leq 0$ → $(- \infty, - 5]$
Write one from its solution set: Write a rational inequality whose solution set is $[0, 1)$ → One answer:
$\frac{x}{x - 1} \leq 0$

Questions about these worksheets

Yes. Every rational inequalities sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Algebra 2 is usually taken around 10th or 11th grade, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the rational inequalities questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 6: rational inequalities, one fraction against a number, a quadratic over a linear factor, a fraction on each side, a squared factor and write one from its solution set. Tick as many as you want and set how many of each, or let it spread them evenly.

Multiplying both sides by the denominator. You do not know its sign, so you do not know whether the inequality flips. Get everything on one side and use the chart instead.

Rational inequalities worksheet for Algebra 2

Select your difficulty

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these rational inequalities worksheets free?+

Do the rational inequalities worksheets come with an answer key?+

Are these rational inequalities worksheets printable?+

What grade are these rational inequalities worksheets for?+

Can I get a different version of the same rational inequalities worksheet?+

Is this a free alternative to Kuta Software for rational inequalities?+

Can I choose how hard the rational inequalities questions are?+

What kinds of question are on a rational inequalities worksheet?+

What do students get wrong with rational inequalities?+