Algebra 2 rational inequalities worksheet

The same sign chart as a polynomial, with one extra rule: the zeros of the bottom split the line as well, and they never belong to the answer no matter which way the inequality points.

Select your difficulty

Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name Rational Inequalities Date Period Solve each inequality. Write the answer in interval notation. $\frac{(x - 1)}{x^{2} - 9 x + 14} \geq 0$ $\frac{(x - 4)}{x^{2} + 3 x + 2} > 0$ $\frac{(x + 6)}{x^{2} + 5 x + 6} \geq 0$ $\frac{(x + 5)}{x^{2} - 36} \leq 0$ $\frac{(x + 6)}{x^{2} + 5 x + 6} \leq 0$ $\frac{(x + 2)}{x^{2} + 3 x - 10} \geq 0$ $\frac{(x + 5)}{x^{2} - 6 x - 7} < 0$ $\frac{(x - 7)}{x^{2} + 5 x - 14} > 0$ $\frac{(x + 3)}{x^{2} - 5 x + 4} \geq 0$ $\frac{(x + 3)}{x^{2} + 2 x - 15} > 0$ $\frac{(x + 1)}{x^{2} + 4 x - 5} \geq 0$ $\frac{(x - 2)}{x^{2} - 8 x + 7} < 0$ $\frac{(x + 6)}{x^{2} + 6 x - 7} \geq 0$ $\frac{(x + 2)}{x^{2} + 11 x + 28} \leq 0$ $\frac{(x - 5)}{x^{2} + 5 x + 6} \leq 0$ $\frac{(x + 3)}{x^{2} - 2 x - 3} < 0$ $\frac{(x + 7)}{x^{2} + x - 12} \geq 0$ $\frac{(x + 5)}{x^{2} + 3 x - 4} \geq 0$ $\frac{(x + 5)}{x^{2} + 3 x - 28} > 0$ $\frac{(x - 1)}{x^{2} - 5 x + 6} \geq 0$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Rational Inequalities

Date Period

Solve each inequality. Write the answer in interval notation.

$\frac{(x - 1)}{x^{2} - 9 x + 14} \geq 0$
$\frac{(x - 4)}{x^{2} + 3 x + 2} > 0$
$\frac{(x + 6)}{x^{2} + 5 x + 6} \geq 0$
$\frac{(x + 5)}{x^{2} - 36} \leq 0$
$\frac{(x + 6)}{x^{2} + 5 x + 6} \leq 0$
$\frac{(x + 2)}{x^{2} + 3 x - 10} \geq 0$
$\frac{(x + 5)}{x^{2} - 6 x - 7} < 0$
$\frac{(x - 7)}{x^{2} + 5 x - 14} > 0$
$\frac{(x + 3)}{x^{2} - 5 x + 4} \geq 0$
$\frac{(x + 3)}{x^{2} + 2 x - 15} > 0$
$\frac{(x + 1)}{x^{2} + 4 x - 5} \geq 0$
$\frac{(x - 2)}{x^{2} - 8 x + 7} < 0$
$\frac{(x + 6)}{x^{2} + 6 x - 7} \geq 0$
$\frac{(x + 2)}{x^{2} + 11 x + 28} \leq 0$
$\frac{(x - 5)}{x^{2} + 5 x + 6} \leq 0$
$\frac{(x + 3)}{x^{2} - 2 x - 3} < 0$
$\frac{(x + 7)}{x^{2} + x - 12} \geq 0$
$\frac{(x + 5)}{x^{2} + 3 x - 4} \geq 0$
$\frac{(x + 5)}{x^{2} + 3 x - 28} > 0$
$\frac{(x - 1)}{x^{2} - 5 x + 6} \geq 0$



Sigma Prep · sigmaprep.io/worksheets Name Rational InequalitiesAnswer keyVersion 1 Date Period Solve each inequality. Write the answer in interval notation. $\frac{(x - 1)}{x^{2} - 9 x + 14} \geq 0$ $[1, 2) \cup (7, \infty)$ $\frac{(x - 4)}{x^{2} + 3 x + 2} > 0$ $(- 2, - 1) \cup (4, \infty)$ $\frac{(x + 6)}{x^{2} + 5 x + 6} \geq 0$ $[- 6, - 3) \cup (- 2, \infty)$ $\frac{(x + 5)}{x^{2} - 36} \leq 0$ $(- \infty, - 6) \cup [- 5, 6)$ $\frac{(x + 6)}{x^{2} + 5 x + 6} \leq 0$ $(- \infty, - 6] \cup (- 3, - 2)$ $\frac{(x + 2)}{x^{2} + 3 x - 10} \geq 0$ $(- 5, - 2] \cup (2, \infty)$ $\frac{(x + 5)}{x^{2} - 6 x - 7} < 0$ $(- \infty, - 5) \cup (- 1, 7)$ $\frac{(x - 7)}{x^{2} + 5 x - 14} > 0$ $(- 7, 2) \cup (7, \infty)$ $\frac{(x + 3)}{x^{2} - 5 x + 4} \geq 0$ $[- 3, 1) \cup (4, \infty)$ $\frac{(x + 3)}{x^{2} + 2 x - 15} > 0$ $(- 5, - 3) \cup (3, \infty)$ $\frac{(x + 1)}{x^{2} + 4 x - 5} \geq 0$ $(- 5, - 1] \cup (1, \infty)$ $\frac{(x - 2)}{x^{2} - 8 x + 7} < 0$ $(- \infty, 1) \cup (2, 7)$ $\frac{(x + 6)}{x^{2} + 6 x - 7} \geq 0$ $(- 7, - 6] \cup (1, \infty)$ $\frac{(x + 2)}{x^{2} + 11 x + 28} \leq 0$ $(- \infty, - 7) \cup (- 4, - 2]$ $\frac{(x - 5)}{x^{2} + 5 x + 6} \leq 0$ $(- \infty, - 3) \cup (- 2, 5]$ $\frac{(x + 3)}{x^{2} - 2 x - 3} < 0$ $(- \infty, - 3) \cup (- 1, 3)$ $\frac{(x + 7)}{x^{2} + x - 12} \geq 0$ $[- 7, - 4) \cup (3, \infty)$ $\frac{(x + 5)}{x^{2} + 3 x - 4} \geq 0$ $[- 5, - 4) \cup (1, \infty)$ $\frac{(x + 5)}{x^{2} + 3 x - 28} > 0$ $(- 7, - 5) \cup (4, \infty)$ $\frac{(x - 1)}{x^{2} - 5 x + 6} \geq 0$ $[1, 2) \cup (3, \infty)$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Rational InequalitiesAnswer keyVersion 1

Date Period

Solve each inequality. Write the answer in interval notation.

$\frac{(x - 1)}{x^{2} - 9 x + 14} \geq 0$ $[1, 2) \cup (7, \infty)$
$\frac{(x - 4)}{x^{2} + 3 x + 2} > 0$ $(- 2, - 1) \cup (4, \infty)$
$\frac{(x + 6)}{x^{2} + 5 x + 6} \geq 0$ $[- 6, - 3) \cup (- 2, \infty)$
$\frac{(x + 5)}{x^{2} - 36} \leq 0$ $(- \infty, - 6) \cup [- 5, 6)$
$\frac{(x + 6)}{x^{2} + 5 x + 6} \leq 0$ $(- \infty, - 6] \cup (- 3, - 2)$
$\frac{(x + 2)}{x^{2} + 3 x - 10} \geq 0$ $(- 5, - 2] \cup (2, \infty)$
$\frac{(x + 5)}{x^{2} - 6 x - 7} < 0$ $(- \infty, - 5) \cup (- 1, 7)$
$\frac{(x - 7)}{x^{2} + 5 x - 14} > 0$ $(- 7, 2) \cup (7, \infty)$
$\frac{(x + 3)}{x^{2} - 5 x + 4} \geq 0$ $[- 3, 1) \cup (4, \infty)$
$\frac{(x + 3)}{x^{2} + 2 x - 15} > 0$ $(- 5, - 3) \cup (3, \infty)$
$\frac{(x + 1)}{x^{2} + 4 x - 5} \geq 0$ $(- 5, - 1] \cup (1, \infty)$
$\frac{(x - 2)}{x^{2} - 8 x + 7} < 0$ $(- \infty, 1) \cup (2, 7)$
$\frac{(x + 6)}{x^{2} + 6 x - 7} \geq 0$ $(- 7, - 6] \cup (1, \infty)$
$\frac{(x + 2)}{x^{2} + 11 x + 28} \leq 0$ $(- \infty, - 7) \cup (- 4, - 2]$
$\frac{(x - 5)}{x^{2} + 5 x + 6} \leq 0$ $(- \infty, - 3) \cup (- 2, 5]$
$\frac{(x + 3)}{x^{2} - 2 x - 3} < 0$ $(- \infty, - 3) \cup (- 1, 3)$
$\frac{(x + 7)}{x^{2} + x - 12} \geq 0$ $[- 7, - 4) \cup (3, \infty)$
$\frac{(x + 5)}{x^{2} + 3 x - 4} \geq 0$ $[- 5, - 4) \cup (1, \infty)$
$\frac{(x + 5)}{x^{2} + 3 x - 28} > 0$ $(- 7, - 5) \cup (4, \infty)$
$\frac{(x - 1)}{x^{2} - 5 x + 6} \geq 0$ $[1, 2) \cup (3, \infty)$

How to do these

$\frac{x - 1}{x + 2} \leq 0$

The chart splits at -2 and 1.
Between them the fraction is negative.
1 is allowed, -2 is not.

The answer is $(- 2, 1]$ .

Where students go wrong

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.

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

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.

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.

Algebra 2 rational inequalities worksheet

Select your difficulty

Rational Inequalities

Rational InequalitiesAnswer keyVersion 1

How to do these

Where students go wrong

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?+

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