GMAT Exam: A Simple Guide To Algebra - One-Variable Linear Equations and Inequalities

Solving One-Variable Linear Equations
To solve a one-variable linear equation, follow these steps.

Step 1. Remove parentheses, if any, using the distributive property.
Step 2. Combine like terms, if any, on each side of the equation.
Step 3. If variable terms are on both sides of the equation, add a variable term to both sides of the equation or subtract a variable term from both sides so that all variable terms are on only one side of the equation. Then simplify.
Step 4. Isolate the variable term. If a number is added to the variable term, subtract that number from both sides of the equation. If a number is subtracted from the variable term, add that number to both sides of the equation. Then simplify.
Step 5. Make the coefficient of the variable 1 by dividing both sides of the equation by the variable’s coefficient.

Tip: If the coefficient is a fraction, do the division by multiplying both sides of the equation by the fraction’s reciprocal. Then simplify.

You can check the solution by substituting it into the original equation.

Here are examples.
Solve –2(1 – x) + 4 x = 3x – 14.

Skip steps that are not needed for the particular equation you are solving.

Solving Two-Variable Linear Equations for One of the Variables
Use the procedure for solving a linear equation in one variable to solve a two-variable linear equation, such as 3x + 2y = –4, for one of the variables in terms of the other variable.

As you solve for the variable of interest, you simply treat the other variable as you would a constant.

For instance, to solve 3x + 2y = –4 for y, do as follows:

.

You must divide both terms of the numerator by –2.

Solving Linear Inequalities
You solve linear inequalities in much the same way you solve linear equations.

There is just one very important difference.

When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol.

Here is an example:

 

You can write the two inequalities, –12 < 5x + 2 and 5x + 2 < 12, as the double inequality –12 < 5x + 2 < 12.

To solve a double inequality, undo what has been done to the variable expression.

Perform the “undoing” operations on all three of the expressions that make up the double inequality.

For example, to solve –12 < 5x + 2 and 5x + 2 < 12, proceed as follows.