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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

Khan Academy video wrapper
Intro to square-root equations & extraneous solutionsSee video transcript

What do I need to know to solve radical equations?

The process of solving radical equations almost always involves rearranging the radical equations into
quadratic equations
, 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:
  1. Isolate the radical expression to one side of the equation.
  2. Square both sides the equation.
  3. 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:
  1. Solve the radical equation as outlined above.
  2. 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!

Try: identify the steps to solving a radical equation
square root of, 2, x, minus, 9, end square root, equals, x, minus, 6
To solve the equation above, we first
both sides of the equation, then rewrite the result as a
equation. Solving this equation gives us 2 solutions, and we
check for extraneous solutions.


Try: Identify an extraneous solution to a radical equation
Marcy solved the radical equation square root of, x, plus, 2, end square root, equals, x and got two solutions, 2 and minus, 1.
When we substitute 2 for x into the equation, the left side of the equation is square root of, 2, plus, 2, end square root, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
and the right side of the equation is 2. Therefore, 2 is
.
When we substitute minus, 1 for x into the equation, the left side of the equation is square root of, minus, 1, plus, 2, end square root, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
and the right side of the equation is minus, 1. Therefore, minus, 1 is
.


How do I solve rational equations?

Equations with rational expressions

Khan Academy video wrapper
Equations with rational expressionsSee video transcript

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:
  1. Rewrite the equation until the variable no longer appears in the denominators of rational expressions.
  2. 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:
  1. Solve the rational equation as outlined above.
  2. 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!

TRY: Identify the steps to solving a rational equation
start fraction, 3, x, divided by, x, plus, 2, end fraction, equals, 2
To solve the equation above, we first
both sides of the equation by x, plus, 2, then solve the resulting
equation.
Because the denominator of the rational expression is x, plus, 2, the only value of x that would lead to division by 0 is
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
. Therefore, when we get 4 as the solution, we know that it is
.


TRY: Identify an extraneous solution to a rational equation
Mehdi solved the rational equation start fraction, x, squared, minus, 4, divided by, x, plus, 2, end fraction, equals, 4 and got two solutions, minus, 2 and 6.
When we substitute minus, 2 for x into the equation, the denominator of the rational expression is minus, 2, plus, 2, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
. Therefore, minus, 2 is
.
When we substitute 6 for x into the equation, the denominator of the rational expression is 6, plus, 2, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
and the rational expression is equal to start fraction, 6, squared, minus, 4, divided by, 6, plus, 2, end fraction, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
. Therefore, 6 is
.


How do I solve absolute value equations?

Absolute value equation with two solutions

Khan Academy video wrapper
Worked example: absolute value equation with two solutionsSee video transcript

Absolute value equation with no solution

Khan Academy video wrapper
Worked example: absolute value equations with no solutionSee video transcript
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!

try: write two linear equations from an absolute value equation
open vertical bar, 3, x, plus, 7, close vertical bar, equals, 16
To solve the absolute value equation above, we must solve two linear equations.
x, equals, 3 is one solution to the absolute value equation and satisfies the linear equation
.
x, equals, minus, start fraction, 23, divided by, 3, end fraction is the other solution to the absolute value equation and satisfies the linear equation
.


Your turn!

Practice: solve a radical equation
square root of, 6, x, plus, 9, end square root, equals, x, plus, 3
Which of the following values of x satisfies the equation above?
Choose 1 answer:
Choose 1 answer:


Practice: check for extraneous solutions to a radical equation
square root of, 4, x, plus, 16, end square root, equals, x, plus, 1
Which of the following are the solutions to the equation above?
I. minus, 3
I, I. 5
Choose 1 answer:
Choose 1 answer:


Practice: solve a rational equation
If start fraction, 1, divided by, x, plus, 1, end fraction, equals, start fraction, 3, divided by, 5, x, minus, 1, end fraction, what is the value x ?
Choose 1 answer:
Choose 1 answer:


Practice: solve a rational equation
Which of the following values of x satisfies the equation start fraction, x, squared, minus, 4, x, divided by, x, minus, 4, end fraction, equals, 1 ?
Choose 1 answer:
Choose 1 answer:


practice: solve an absolute value equation
open vertical bar, x, minus, 7, close vertical bar, equals, 1
If a and b are solutions to the equation above, what is the value of a, plus, b ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text


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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