## What are radical, rational, and absolute value equations?

**Radical equations** are equations in which variables appear under radical symbols (

is a radical equation.$\sqrt{2x-1}=x$

**Rational equations** are equations in which variables can be found in the denominators of rational expressions.

is a rational equation.$\frac{1}{x+1}}={\displaystyle \frac{2}{x}$

Both radical and rational equations can have **extraneous solutions**, algebraic solutions that emerge as we solve the equations that do not satisfy the original equations. In other words, extraneous solutions *seem* like they're solutions, but they aren't.

**Absolute value equations** are equations in which variables appear within vertical bars (

is an absolute value equation.$|x+1|=2$

In this lesson, we'll learn to:

- Solve radical and rational equations
- Identify extraneous solutions to radical and rational equations
- Solve absolute value equations

**You can learn anything. Let's do this!**

## How do I solve radical equations?

### Intro to square-root equations & extraneous solutions

Khan Academy video wrapper

Intro to square-root equations & extraneous solutions

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### What do I need to know to solve radical equations?

The process of solving radical equations almost always involves rearranging the radical equations into

, then solving the quadratic equations. As such, knowledge of how to manipulate polynomials algebraically and solve a variety of quadratic equations is essential to successfully solving radical equations.

To solve a radical equation:

- Isolate the radical expression to one side of the equation.
- Square both sides the equation.
- Rearrange and solve the resulting equation.

**Example:** If

We can factor

$a+b=-2$ $ab=1$

$-1+(-1)=-2$ $(-1)(-1)=1$

Now, we can solve for

When it comes to extraneous solutions, the concept that confuses the most students is that of the **principal square root**. The square root operation gives us only the principal square root, or positive positive square root. For example, *not* both

In most cases, solving radical equations on the SAT involves squaring both sides of the radical equation. Raising both sides of an equation to an *even power* is not a *reversible* operation. For example, if

Let's look at a numerical example. For *must* perform additional checks to make sure that

To check for extraneous solutions to a radical equation:

- Solve the radical equation as outlined above.
- Substitute the solutions into the original equation. A solution is extraneous if it does not satisfy the original equation.

**Example:** What is the solution to the equation

We can factor

$a+b=-3$ $ab=-4$

$-4+1=-3$ $(-4)(1)=-4$

Now, we can solve for

Now, we need to substitute

### Try it!

Try: identify the steps to solving a radical equation

To solve the equation above, we first

both sides of the equation, then rewrite the result as a

equation. Solving this equation gives us

check for extraneous solutions.

Try: Identify an extraneous solution to a radical equation

Marcy solved the radical equation

When we substitute

and the right side of the equation is

.

When we substitute

and the right side of the equation is

.

## How do I solve rational equations?

### Equations with rational expressions

Khan Academy video wrapper

Equations with rational expressions

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### What do I need to know to solve rational equations?

Knowledge of fractions, polynomial operations and factoring, and quadratic equations is essential for successfully solving rational equations.

To solve a rational equation:

- Rewrite the equation until the variable no longer appears in the denominators of rational expressions.
- Rearrange and solve the resulting linear or quadratic equation.

**Example:** If

Most often, the reason a solution to a rational equation is extraneous is because the solution, when substituted into the original equation, results in division by

To check for extraneous solutions to a rational equation:

- Solve the rational equation as outlined above.
- Substitute the solution(s) into the original equation. A solution is extraneous if it does not satisfy the original equation.

**Example:** What value(s) of

However, when we substitute

**No value of **

### Try it!

TRY: Identify the steps to solving a rational equation

To solve the equation above, we first

both sides of the equation by

equation.

Because the denominator of the rational expression is

. Therefore, when we get

.

TRY: Identify an extraneous solution to a rational equation

Mehdi solved the rational equation

When we substitute

. Therefore,

.

When we substitute

and the rational expression is equal to

. Therefore,

.

## How do I solve absolute value equations?

### Absolute value equation with two solutions

Khan Academy video wrapper

Worked example: absolute value equation with two solutions

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### Absolute value equation with no solution

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Worked example: absolute value equations with no solution

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The absolute value of a number is equal to the number's *distance* from *always positive*. For example:

- The absolute value of
, or$2$ , is$|2|$ .$2$ - The absolute value of
, or$-2$ , is also$|-2|$ .$2$

Practically, this means every absolute value equation can be split into two linear equations. For example, if

- The absolute value equation is true if
.$2x+1=5$ - The absolute value equation is
*also*true if since$2x+1=-5$ .$|-5|=5$

When solving absolute value equations, rewrite the equation as two linear equations, then solve each linear equation. Both solutions are solutions to the absolute value equation.

**Example:** What are the solutions to the equation

The absolute value equation can be divided into two linear equations:

**The solutions are **

### Try it!

try: write two linear equations from an absolute value equation

To solve the absolute value equation above, we must solve two linear equations.

.

.

## Your turn!

Practice: solve a radical equation

Which of the following values of

Practice: check for extraneous solutions to a radical equation

Which of the following are the solutions to the equation above?

Practice: solve a rational equation

If

Practice: solve a rational equation

Which of the following values of

practice: solve an absolute value equation

If

## Things to remember

The radical operator (*positive* square root. If a solution leads to equating the square root of a number to a negative number, then that solution is extraneous.

We cannot divide by

For the absolute value equation

$ax+b=c$ $ax+b=-c$

Both solutions are solutions to the absolute value equation.