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### Course: Digital SAT Math > Unit 8

Lesson 9: Radical, rational, and absolute value equations: medium# Radical, rational, and absolute value equations | Lesson

A guide to radical, rational, and absolute value equations on the digital SAT

## 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

### 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

When it comes to extraneous solutions, the concept that confuses the most students is that of the $\sqrt{4}=2$ , $-2$ and $2$ even though $(-2{)}^{2}={2}^{2}=4$ . If a solution leads to equating the square root of a number to a negative number, then that solution is extraneous.

**principal square root**. The square root operation gives us only the principal square root, or positive positive square root. For example,*not*bothTo 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

### Try it!

## How do I solve rational equations?

### Equations with rational expressions

### 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 $0$ . For example, if one of the solutions to a rational equation is $2$ and the original equation contains the denominator $x-2$ , then the solution $2$ is extraneous because $2-2=0$ , and we cannot divide by $0$ .

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

### Try it!

## How do I solve absolute value equations?

### Absolute value equation with two solutions

### Absolute value equation with no solution

The absolute value of a number is equal to the number's $0$ on the number line, which means the absolute value of a nonzero number is

*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 $|2x+1|=5$ :

- 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

### Try it!

## Your turn!

## Things to remember

The radical operator ($\sqrt{\phantom{x}}$ ) calculates only the

*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 $0$ . If a solution leads to division by $0$ , then that solution is extraneous.

For the absolute value equation $|ax+b|=c$ , rewrite the equation as the following linear equations and solve them.

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

Both solutions are solutions to the absolute value equation.

## Want to join the conversation?

- Upvote if you think Sal rocks 🤘🏽(361 votes)
- I did 👍
**All for Sal, and Sal for all**!(28 votes)

- in a few years this comment section will be filled and we will become history(88 votes)
- What date are you guys doing the digital SAT?(3 votes)

- oct 7 2023.....

pray for me(33 votes)- I take it on August 26th :/(19 votes)

- Gonna be waiting for "how's life bro?" questions in 6 years xd(11 votes)
- early but hows life bro(6 votes)

- been studying for the SAT i have saturday (3/11/23) ill let you guys know how i did(2 votes)
- he lied....(24 votes)

- This lesson tells us to take only the positive value out of a square root whereas the lesson before this(solving quadratic equations)tells us to take both the positive and the negative value. How do I know when to follow which lesson on the test?(10 votes)
- Equations such as x^2=a will have both positive and negative solutions. But if you are using the
**square root function**with this symbol "√" then we only take the principal or positive square root. The square root operation is only supposed to give you the positive square root. So for equations like √x=a, only take the positive solution.(3 votes)

- 9 March SAT => 1500+ wait for me(6 votes)
- how'd it go blud(1 vote)

- easiest way to get the solutions and correct ones every time is to just plot the initial equations on desmos no?(5 votes)
- can someone explain how sqrt(8t) can also be written or simplified to 2*sqrt(2t) ?(3 votes)
- one of the rules of roots is that sqrt(a * b) = sqrt(a) * sqrt(b)

sqrt(8t) = sqrt(4 * 2t) = sqrt(4) * sqrt(2t) = 2sqrt(2t).(4 votes)