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## Digital SAT Math

### Unit 12: Lesson 9

Radical, rational, and absolute value equations: advanced# 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 (square root of, empty space, end square root).

- square root of, 2, x, minus, 1, end square root, equals, x is a radical equation.

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

- start fraction, 1, divided by, x, plus, 1, end fraction, equals, start fraction, 2, divided by, x, end fraction is a rational equation.

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 (open vertical bar, close vertical bar).

- open vertical bar, x, plus, 1, close vertical bar, equals, 2 is an absolute value equation.

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 square root of, 2, x, minus, 1, end square root, equals, x, what is the value of x ?

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, square root of, 4, end square root, equals, 2,*not*both minus, 2 and 2 even though left parenthesis, minus, 2, right parenthesis, squared, equals, 2, squared, equals, 4. If a solution leads to equating the square root of a number to a negative number, then that solution is extraneous.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 square root of, 3, x, plus, 4, end square root, equals, x ?

### 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 start fraction, 1, divided by, x, plus, 1, end fraction, equals, start fraction, 2, divided by, x, end fraction, what is the value of x ?

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, minus, 2, then the solution 2 is extraneous because 2, minus, 2, equals, 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 x satisfies the equation start fraction, 2, divided by, x, minus, 1, end fraction, equals, start fraction, x, plus, 1, divided by, x, minus, 1, end fraction ?

### 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

*distance*from 0 on the number line, which means the absolute value of a nonzero number is*always positive*. For example:- The absolute value of 2, or open vertical bar, 2, close vertical bar, is 2.
- The absolute value of minus, 2, or open vertical bar, minus, 2, close vertical bar, is also 2.

Practically, this means every absolute value equation can be split into two linear equations. For example, if open vertical bar, 2, x, plus, 1, close vertical bar, equals, 5:

- The absolute value equation is true if 2, x, plus, 1, equals, 5.
- The absolute value equation is
*also*true if 2, x, plus, 1, equals, minus, 5 since open vertical bar, minus, 5, close vertical bar, equals, 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 open vertical bar, 2, x, minus, 1, close vertical bar, equals, 5 ?

### Try it!

## Your turn!

## Things to remember

The radical operator (square root of, empty space, end square root) 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 open vertical bar, a, x, plus, b, close vertical bar, equals, c, rewrite the equation as the following linear equations and solve them.

- a, x, plus, b, equals, c
- a, x, plus, b, equals, minus, c

Both solutions are solutions to the absolute value equation.

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