Equations and Inequalities

• First-Degree Equations and Inequalities

• Absolute Value, Radical, and Fractional Equations and Inequalities

• Quadratic Equations

• Exponential Equations

• Systems of Linear Equations

• The Addition Method

• The Substitution Method

• The Graphing Method

• Solving Linear-Quadratic Systems

• Exercises

• Answers Explained

On a typical Math 1 test, several questions require you to solve equations and inequalities. In this section, we will review the different types of equations and inequalities you will encounter and the methods for solving them.

The basic principle that you must adhere to in solving any equation is that you can manipulate it in any way as long as you do the same thing to both sides. For example, you may add the same number to each side, subtract the same number from each side, multiply or divide each side by the same number (except 0), square each side, take the square root of each side (if the quantities are positive), take the reciprocal of each side, take the logarithm of each side, and so on. These comments apply to inequalities as well. However, you must be very careful when working with inequalities because some procedures, such as multiplying or dividing by a negative number and taking reciprocals, reverse inequalities.


The simplest equations and inequalities that you will have to solve on the Math 1 test have only one variable and no exponents. They are called first-degree or linear equations and inequalities. You can always use the six-step method described below to solve them.

EXAMPLE 1: The following solution of the equation

 illustrates each of the six steps.


What You Should Do



Get rid of fractions (and decimals) by multiplying both sides by a common denominator.

Multiply each term on both sides of the equation by 2: x + 6(x – 2) = 4(x + 1) + 2


Get rid of all parentheses by using the distributive law.

x + 6x – 12 = 4x + 4 + 2


Combine like terms on each side.

7x – 12 = 4x + 6


By adding or subtracting, get all the variables on one side.

Subtract 4x from each side: 3x – 12 = 6


By adding or subtracting, get all the plain numbers onto the other side.

Add 12 to each side: 3x = 18


Divide both sides by the coefficient of the variable.*

Divide both sides by 3: x = 6

*NOTE: If you use this method on an inequality and in Step 6 you divide by a negative number, remember to reverse the inequality.

The example worked out above is unusual because it required all six steps. When you solve an equation on the Math 1 test, you will probably not need to do every step. Think of the six steps as a list of questions that must be answered in that order. Ask if each step is necessary. If it isn’t, move on to the next one; if it is necessary, do it. Example 2, below, does not require all six steps.

EXAMPLE 2: For what value of x is 5(x – 10) = x + 10?


Memorize these six steps in order, and use this method whenever you have to solve this type of equation or inequality.

When you use the six-step method, do not actually write out a table. Rather, use the method as a guideline and mentally go through each step, doing whichever ones are required. From now on, we will just do each step, using the symbol ⇒ to move from one step to the next.

Sometimes an equation on the Math 1 test contains several variables and you have to solve for one of them in terms of the others.


When you have to solve for one variable in terms of others, treat all the others as if they were numbers, and apply the six-step method.

EXAMPLE 3: If , what is the value of y in terms of x and z?

Multiply both sides by 3:     

Add 12z to each side:     2x + 12z = 9y 

Divide both sides by 9:          

EXAMPLE 4: If v is an integer and the average (arithmetic mean) of 1, 2, 3, and v is less than 20, what is the greatest possible value of v ?

Write the inequality  and use the six-step method:

Since v is an integer, the greatest possible value of v is 73.