# Equations and Inequalities

## Other Types of Equations

We have solved linear equations, rational equations, and quadratic equations using several methods. However, there are many other types of equations, and we will investigate a few more types in this section. We will look at equations involving rational exponents, polynomial equations, radical equations, absolute value equations, equations in quadratic form, and some rational equations that can be transformed into quadratics. Solving any equation, however, employs the same basic algebraic rules. We will learn some new techniques as they apply to certain equations, but the algebra never changes.

### Solving Equations Involving Rational Exponents

Rational exponents are exponents that are fractions, where the numerator is a power and the denominator is a root. For example, 
  is another way of writing 
  
  is another way of writing 
  The ability to work with rational exponents is a useful skill, as it is highly applicable in calculus.
We can solve equations in which a variable is raised to a rational exponent by raising both sides of the equation to the reciprocal of the exponent. The reason we raise the equation to the reciprocal of the exponent is because we want to eliminate the exponent on the variable term, and a number multiplied by its reciprocal equals 1. For example, 
  
  and so on.

### Solving Equations Using Factoring

We have used factoring to solve quadratic equations, but it is a technique that we can use with many types of polynomial equations, which are equations that contain a string of terms including numerical coefficients and variables. When we are faced with an equation containing polynomials of degree higher than 2, we can often solve them by factoring.

### Solving Radical Equations

Radical equations are equations that contain variables in the radicand (the expression under a radical symbol), such as
Radical equations may have one or more radical terms, and are solved by eliminating each radical, one at a time. We have to be careful when solving radical equations, as it is not unusual to find extraneous solutions, roots that are not, in fact, solutions to the equation. These solutions are not due to a mistake in the solving method, but result from the process of raising both sides of an equation to a power. However, checking each answer in the original equation will confirm the true solutions.

### Solving an Absolute Value Equation

Next, we will learn how to solve an absolute value equation. To solve an equation such as 
  we notice that the absolute value will be equal to 8 if the quantity inside the absolute value bars is 
  or 
  This leads to two different equations we can solve independently.
Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.

### Solving Other Types of Equations

There are many other types of equations in addition to the ones we have discussed so far. We will see more of them throughout the text. Here, we will discuss equations that are in quadratic form, and rational equations that result in a quadratic.

### Solving Equations in Quadratic Form

Equations in quadratic form are equations with three terms. The first term has a power other than 2. The middle term has an exponent that is one-half the exponent of the leading term. The third term is a constant. We can solve equations in this form as if they were quadratic. A few examples of these equations include 
  and 
  In each one, doubling the exponent of the middle term equals the exponent on the leading term. We can solve these equations by substituting a variable for the middle term.

### Solving Rational Equations Resulting in a Quadratic

Earlier, we solved rational equations. Sometimes, solving a rational equation results in a quadratic. When this happens, we continue the solution by simplifying the quadratic equation by one of the methods we have seen. It may turn out that there is no solution.

### Key Concepts

1. Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1. See , , and .
2. Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping. See  and .
3. We can solve radical equations by isolating the radical and raising both sides of the equation to a power that matches the index. See  and .
4. To solve absolute value equations, we need to write two equations, one for the positive value and one for the negative value. See .
5. Equations in quadratic form are easy to spot, as the exponent on the first term is double the exponent on the second term and the third term is a constant. We may also see a binomial in place of the single variable. We use substitution to solve. See  and .
6. Solving a rational equation may also lead to a quadratic equation or an equation in quadratic form. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, solve the rational exponent equation. Use factoring where necessary.
For the following exercises, solve the following polynomial equations by grouping and factoring.
For the following exercises, solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions.
For the following exercises, solve the equation involving absolute value.
For the following exercises, solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring.

### Extensions

For the following exercises, solve for the unknown variable.

### Real-World Applications

For the following exercises, use the model for the period of a pendulum, 
  such that 
  where the length of the pendulum is L and the acceleration due to gravity is 
  
For the following exercises, use a model for body surface area, BSA, such that 
  where w = weight in kg and h = height in cm.