PAM - Rational and Radical Equations Lesson

Rational and Radical Equations

What is a rational number?

A rational number is any number that can be written as a fraction. These include 2, 700, 1/3, 7/8, etc. Note that 2 and 700 can both be written as fractions, they are 2/1 and 700/1.

What is a radical number?

A radical number is a number that is written with the radical sign, √, or with fractional exponents. As a decimal the number could be repeating. Examples are √2, √11, equation image indicator . Like square roots, these can be radicals with roots other than 2.

What is a rational exponent?

A rational number as an exponent. The fractional portion of the exponent consists of the multiplication power/radical root. For example, an exponent of 2/3 means to multiply the base of the exponent two times and take the cube root of the answer or take the cube root of the base of the exponent and then multiply twice.

Review

Domain - all of the x-values that may be put into the equation to create y-values. For lines, parabolas, and exponential functions, this is all real numbers.

Range - all of the y-values that can be created when all of the x-values are put in equation. For lines, this is all real numbers. For parabolas and exponentials this is restricted based on the location of the graph.

Function - an equation which is intersected at one point anywhere on the graph, wherever a vertical line is drawn.

Solving Rational Functions

What does a rational function look like? Here is the parent rational function. All others are similar, just moved around on the graph. The parent function $LaTeX: f\left(x\right)=\frac{1}{x}$ $f\left(x\right)=\frac{1}{x}$ has two parts, one in the 1^st quadrant and one in the 3^rd quadrant.

image graph Here is the parent rational function. All others are similar, just moved around on the graph. The parent function f(x) = 1/x has two parts, one in the 1st quadrant and one in the 3rd quadrant.

The parts are separated by two asymptotes, x = 0 and y = 0. Looking at the equation, it makes sense that x = 0 is a vertical asymptote, division by 0 is illegal, so there could never be a y answer for x = 0.

The second asymptote, y = 0, the horizontal asymptote requires some reasoning.

Is there a way to get 1/x, where x may be any number except 0, to = 0?

No, as the number always divides into 1.

The number will be a very large number as the y-axis is approached with x-values. For example, 1/ (1/10) = 1 * 10/1 = 10, 1/ (1/100) = 1 * 100/1 = 100, etc. The values of y continue to get larger as x gets closer to 0, since the fractions in the denominator are inverted and multiplied.

All rational functions have two parts to their graph. The difference is that the vertical asymptote moves left and right based on the x expression, x ± #, in the denominator. We will see this in examples that follow.

Domain of Rational Functions

Consider $LaTeX: f\left(x\right)=\frac{1}{x}$ $f\left(x\right)=\frac{1}{x}$ . The domain of this function is all real numbers, but x cannot be equal to 0. This is written as all reals, x ≠ 0.

Consider $LaTeX: f\left(x\right)=\frac{1}{x+2}$ $f\left(x\right)=\frac{1}{x+2}$ . The domain of this function is all real numbers, but x cannot be equal to 2. This is written as all reals, x ≠ -2. Why not equal to -2? Because if x is = to -2 then x + 2 = -2 + 2 = 0 and the denominator would contain 0 and we cannot divide by zero.

So the domain changes from all reals to reals with a restriction for rational functions. To find the restriction, simply take the denominator and set it not equal to 0 and solve for x.

x + 2 ≠ 0

x ≠ −2 , so x = −2 is the vertical asymptote. Note this on the graph below.

image of graph The horizontal asymptote remains y = 0 unless we add the + c after the basic equation to move the graph up and down. The + c moves all graphs up and down.

The horizontal asymptote remains y = 0 unless we add the + c after the basic equation to move the graph up and down. The + c moves all graphs up and down.

Solve Rational Functions

Now it's time to examine how to solve rational functions. A great way for you to do this part would be for you to copy the problem and try it, then look at the explanation to assist with more detailed understanding.

Note that some rational functions can be constructed as a proportion that we studied earlier, a fraction on each side. Creating a fraction by using division by 1 is perfectly acceptable. However, you cannot use the cross multiply technique if there is anything else besides exactly one fraction on each side. All of the first set may be set up as cross multiplication which allows us to multiply to both sides by both divisors at the same time.

Now copy each problem and work on your own, then review the steps here. Note that some steps could be combined, but all steps are shown so that the detail is available.

Example 1: Solve the function equation image indicator = 10. State the domain.

= 10 Domain is all reals, x ≠ 0, a vertical asymptote at x = 0

= Create both sides as a fraction. One as a denominator always creates a fraction.

10x = 1 * 1 Cross multiply (allowed with an = sign). This is the same as multiplying both sides by x to move x to the numerator. Cross multiplication can only take place if there is one fraction on both sides.

x = Solved. Substitute in the original equation for x to check.

= 1 / = 1 * = 10

Okay, a great basic beginning for starting. Notice that you can check your work. Always work one side of the equation to get the answer on the other when checking. Do no cross the equal sign. Now let's add a longer expression in x.

Example 2: Solve = 3. State the domain.

$LaTeX: \frac{5}{x+2}=3$ $\frac{5}{x+2}=3$ Domain all reals, x ≠ -2, a vertical asymptote

$LaTeX: \frac{5}{x+2}=\frac{3}{1}$ $\frac{5}{x+2}=\frac{3}{1}$ Set up for cross multiplication.

3(x + 2) = 5 * 1 Cross multiply since there was only one fraction on both sides.

x + 2 = Divide by 3.

x = - 2 Subtract

x = - Get common denominator.

x = - Combine like terms.

Check.

= 5/ (- + ) Common denominator

= 5/ Division by a fraction

= 5 * Multiply by the reciprocal fraction

= 3 = 1

Common denominators and multiplying by the reciprocal

For the common denominator, and how to handle a great technique to remember is to remove the fraction from the denominator. Many times we did this when we checked our work! Now let's try a more difficult one with a larger expression in x.

Example 3: Solve Domain all reals x, x ≠

Reason: 2x - 3 ≠ 0, so x = is an excluded value (a value that is excluded from the domain).

6(2x - 3) = 3 * 5 Cross multiply

2x - 3 = equation image indicator Divide by 6

2x - 3 = Factor 6 to reduce

2x - 3 = Simplify = 1

2x = + 3 Add 3

2x = + Common denominator

2x = Add numerators

x = Divide by 2 = multiply by ½

x = Solved

Check: $LaTeX: \frac{3}{2\left(\frac{11}{4}\right)-3}=\frac{3}{\frac{11}{2}-3}$ $\frac{3}{2\left(\frac{11}{4}\right)-3}=\frac{3}{\frac{11}{2}-3}$ Simplify

= Common denominator

= Add numerators

= 3 * Multiply by the reciprocal

The above examples allowed us to review techniques of common denominators, multiplication by the reciprocal, and adding or subtraction fractions. Only one variable was involved in the equations.

Watch the videos below for some more examples and techniques with rational equations including what to do if you have two radicals in an equation.

Now let's look at rational equations involving multiple x's. These equations involve the technique of factoring or taking a square root to solve.

Example 4: Solve = x - 8 State the domain.

= x - 8 Domain is all reals, x ≠ 0.

Group the numerator and realize that 1 is the denominator.

x(x - 8) = 1 Cross multiply.

Distribute

Put the equation in general form.

x = - (-8) + or minus the square roots of ((-8)^2 - 4(1)(-1)) / 2(1) Use the quadratic formula because this is not factorable easily.

Simplify.

Factor square root for squares.

Simplify square root.

Factor out 2 from both terms.

x = 4 ± √17 The 2 divides out = 1

So the roots, the zeros, of the equation are 4 + √17 and 4 - √17.

Sometimes we want to be able to factor quadratic equations like , but they are not factorable by traditional techniques.

To write these as factors group them with ( ).

Group x-intercepts when there is more than one.

= 0 = 0

Solve for zero keeping the group together.

The factored equation.

It's time for you to try a few of these to make sure that you understand the techniques prior to going forward. To see detailed solutions, check this handout Links to an external site..

Solving Radical Functions

As we see, the square root function also follows the rules that we have used for linear, exponentials, and quadratics for shifting horizontally and vertically on the graph. These rules also hold for all root functions, third root, fourth root, and so forth, as well as other powers besides quadratics, as we note using the exponent of 1/2. These rule concepts will allow you to reason and make conclusions with all types of graphs in the future.

In the video below you will find additional explanations of radical graphs.

Solve Radical Functions

Example 5. Solve

In order to solve this equation, the square root must be undone. Note that the y-value is equal to 5 and we are looking for this time to solve for the x-value.

= (5)² Square both sides to undo the square root.

x = 25 The square root of 25 is 5.

Check your work: = 5 Substitute 25 for x.

5 = 5

This answer is the positive answer since that was what the original equation was looking for.

Example 6: Solve

Note that the complete expression x - 3 is under the square root symbol.

= (5)² Square both sides.

x - 3 = 25 Squaring a square root always has an answer of what was inside.

x = 28 Add 3 to both sides to yield the answer.

Check your work: = 5 Note: Work each side separately to check.

5 = 5

Both sides match.

Example 7: Solve

√(x + 11))² = (-6)² Square both sides.

x + 11 = 36 The square of the square root side is the inside of the square root. A negative squared is a positive.

x = 25 Simplify by subtracting 11 from both sides.

Check your work:

Substitute in x = 25.

Combine like terms.

6 ≠ -6

Red X

No, this is not correct.

This is when knowing the domain and range come in handy.

The function y = has a domain of x ≥ −11 and a range y ≥ 0.

The function y = -6 has a domain of all real numbers and a range of y = -6.

There is not an intersection of the y's to create an intersection that would be a solution to both sides of the equation.

y = -6

image graph of y=-6.png

Important: Always check your answers for extraneous ones when dealing with even roots or exponents! Extraneous solutions are solutions that you can find algebraically but do not work when put back in the equations.

In the next lesson, we will examine graphs that intersect and how to find the intersections.

View the video below to gain more practice solving radical functions.

Watch this second video to learn some additional techniques for radical graphs including how to handle two radicals in the same equation.

Radical Functions Practice

Now it is your turn to try on your own. Try the following questions to test the knowledge that you have learned. Download the handout Links to an external site. to see detailed solutions.

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