解有理不等式

A rational number is any number or quantity that can be written as a fraction. This includes numbers that look like fractions right away (like $\frac{1}{4}$, for example) and numbers that are fractions in disguise (like $8$, which is a reduced form of the fraction $\frac{8}{1}$).

Examples of rational numbers are: $7$, -$\frac{3}{2}$, $\frac12$

So, a rational inequality is an expression that includes rational numbers and any of the inequality signs ($<$, $>$, $\le$, $\ge$).

An example of a rational inequality is:

$$\frac{3x}{5} > 8$$

The left side of the inequality has the rational component, while the right hand side has a constant value.

When solving rational inequalities, there are four steps we need to take:

1. Rearrange the inequality so there's a single fraction on one side and $0$ on the other

2. Determine what values make the top and bottom of the fraction equal $0$

3. Construct a number line

4. Test points

Let's start with this example:

$$\frac{3x - 6}{x - 2} \ge \frac{x + 2}{x - 2}$$

This example is a rational inequality because

• $≥$ is used to compare values
• Fractions are included

Step 1: Rearrange the inequality so there's a single fraction on one side and $0$ on the other

Do this just like you normally would - add or subtract values from both sides of the expression until you get the format you want.

$$\begin{align} \frac{3x - 6}{x - 2} &\ge \frac{x + 2}{x - 2}\\\ \\\ \frac{3x - 6}{x - 2} -\frac{x + 2}{x - 2} &\ge \frac{x + 2}{x - 2} - \frac{x + 2}{x - 2}\\\ \\\ \frac{2x - 8}{x - 2} &\ge 0 \end{align}$$

Step 2: Determine what values make the top and bottom of the fraction equal $0$

If we plug $x=4$ into the numerator:

$$\begin{align} 2x-8 &= 0 \\ 2x &= 8 \\ x &= 4 \\ \end{align}$$

If we plug in the value $x=2$ into the denominator:

$$\begin{align} x-2 &= 0 \\ x &= 2 \\ \end{align}$$

These two values ($x=4$ and $x=2$) are called your critical values.

Step 3: Construct a number line

We want to construct a number line that includes our two critical values. Let's take a look at the number line and then go over how we got there.

Recall that an open circle on a number line means that point is not included in the range of possible values, while a closed circle means that the point is included.

No matter what, the critical value that makes the bottom of your fraction equal $0$ will have an open circle on the number line. In our case, this is $x=2$. This value makes the denominator zero, making the whole expression undefined. (If you want to know why, ask Siri what happens when you divide by zero).

Now, here's where we need to pay special attention to inequality sign.

• If you're using a plain old $<$ or $>$ sign, the other critical value will have an open circle on the number line.
• If you're using $≤$ or $≥$, the critical value will have a closed circle.

This should make sense, since those signs include the phrase "OR equal to." That means that the solution could be equal to the critical value, so we need to give it a closed circle and include it in our range.

Step 4: Test values

So far, we only know what happens when we plug either of our two critical values into the expression. For us, if we plug in $x=2$, the expression becomes undefined and if we plug in $x=4$, the expression equals $0$.

But we need to figure out what happens when we plug in values from other parts of the number line. To do this, we'll use test values.

We'll need one test value from each "segment" of our number line:

In our case, we need three test values. We've selected $0$, $3$, and $5$, but any number in each segment will do. That means, instead of $0$, you could've picked $-78$; any number less than $2$ will work. The same idea is true for the other segments.

What we need to do now is actually TEST our test values. We'll do this by plugging each one into our inequality and seeing if the result produced can be described as "true."

Let's try this with our first test value, $0$.

$$\begin{align} \frac{2x - 8}{x - 2} \ge 0\\\ \\\ \frac{2(0) - 8}{(0) - 2} \ge 0\\\ \\\ \frac{-8}{-2} \ge 0\\\ \\ 4 \ge 0 \end{align}$$

This is "true," since $4$ is in fact greater than or equal to $0$. So, we need to include this part of the number line in our solution.

We can write it like this: $(-∞,2)$.

This covers every possible value less than (but not including - remember it's an open circle) $2$.

Try plugging in the other two test values and then check what we've got below!

Okay, so let's see what we've got here.

The test value $3$ doesn't produce a true statement (on the left in the image above), since $-2$ is NOT greater than or equal to $0$. This means that no values in that segment of the number line are solutions to our inequality.

The test value $5$ produces a true statement (on the right), since $\frac{2}{3}$ is greater than or equal to $0$. So, we need to include this range in our solution.

We can write it like this: $[4, ∞)$.

This covers every value from and including $4$ (notice the bracket instead of the normal parenthesis, and recall the closed circle on the number line) all the way to infinity.

So, our final answer needs to show all of the ranges of solutions we identified. We can write it like this:

$(-∞,2) ∪ [4, ∞)$

(Video) Solving Rational Inequalities

by Mario's Math Tutoring

In this video by Mario's Math Tutoring, you will walk through an example of how to solve rational inequalities.

Summary

Example #1: Solve the following rational inequality: $x^2-x-12>0$

The first step when solving rational inequalities is to factor. We can factor the left side and get,

$$(x-4)(x+3) > 0$$

Then, you want to set these factors equal to $0$,

$$x-4>0 \rightarrow x>4$$ $$x+3 >0 \rightarrow x>-3$$

It's often easier to draw out a number line to understand what the solutions can be. Once we do that and test some points, we find that the solution can be written as: $(-\infty, -3) \bigcup (4,\infty)$. So, any point in those regions would make the rational inequality true.

Try the next two examples on your own before watching Mario!

Example #2: $\frac{x^2-x-2}{x-3} \leq 0$

Now, try the next example.

Example #3: $\frac{3}{x+4} > \frac{2}{x-4}$

Do this on your own before watching the video.

Solving Rational Inequalities

A rational inequality is like a fraction with a variable in it, but instead of an equals sign, it has symbols like $<$, $>$, $\geq$, and $\leq$.

We will use some of the same principles we did to solve rational equations.

There are three major steps you can use to follow to solve rational inequalities.

Step 1: Rearrange so all terms are on one side

Move all the terms to one side of the inequality so that the other side is just $0$. For example:

$$\frac{x+1}{x-2} > 3 \text{ becomes } \frac{x+1}{x-2} -3 >0$$

Then, find a common denominator and simplify the expression into a single rational expression.

Step 2: Find the zeros of the numerator and denominator

Now take your simplified expression. Set the numerator equal to zero to find the solution points - these are the solution points. You'll do the same thing with the denominator, but this will give you the excluded points (where the expression is undefined).

For example, if you have $\frac{x-4}{x+2} > 0$, then:

• $x-4=0 \rightarrow x=4$
• $x+2=0 \rightarrow x=-2$

Step 3: Draw a number line and test points

Plot the numbers you found on a number line. These break the line into sections. Pick a test number from each section and plug it into the simplified expression. Check if the result is positive or negative.

If the inequality says "$>0$", you're looking for where the expression is positive. Use open circles on points that make the denominator $0$ (since they're not allowed) and closed circles if the points is included (like if the sign is $\geq$ or $\leq$).

Looking further

In the next few lessons, you will learn more advanced topics that build on this one. You'll end this set of lessons with partial fraction decomposition!