Equations with One Variable

A two-step linear equation is in the form ax + b = c, where a is a non-zero constant and b and c are constants. There are two basic steps in solving this equation.

STEP BY STEP

Two-Step Linear Equations Review

Multi-Step Linear Equations

In these basic examples of linear equations, the solution may be evident, but these properties demonstrate how to use an opposite operation to solve for a variable. Using these properties, there are three steps in solving a complex linear equation.

STEP BY STEP

Step 1. Simplify each side of the equation. This includes removing parentheses, removing fractions, and adding like terms.

Step 2. Use addition and subtraction properties of an equation to move the variable to one side of the equation and all number terms to the other side of the equation.

Step 3. Use multiplication and division properties of an equation to remove the value in front of the variable.

In Step 2, all of the variables may be placed on the left side or the right side of the equation. The examples in this lesson will place all of the variables on the left side of the equation.

When solving for a variable, apply the same steps as above. In this case, the equation is not being solved for a value, but for a specific variable.

Multi-Step Linear Equations Review

Solving Linear Inequalities

A linear inequality is similar to a linear equation, but it contains an inequality sign (<, >, ≤, ≥). Many of the steps for solving linear inequalities are the same as for solving linear equations. The major difference is that the solution is an infinite number of values. There are four properties to help solve a linear inequality.

Addition Property of Inequality: Add the same number to both sides of the inequality.

Example:
x – 3 < 9
x – 3 + 3 < 9 + 3
x < 12

Subtraction Property of Inequality: Subtract the same number from both sides of the inequality.

Example:
x + 3 > 9
x + 3 – 3 > 9 – 3
x > 6

Multiplication Property of Inequality (when multiplying by a positive number): Multiply both sides of the inequality by the same number.

Example:
\(\frac{x}{3}\) ≥ 9
\(\frac{x}{3} \hspace{0.5mm}\times3\hspace{0.5mm}≥\hspace{0.5mm}9\hspace{0.5mm}\times\hspace{0.5mm}3\)
x ≥ 27

Division Property of Inequality (when multiplying by a positive number): Divide both sides of the inequality by the same number.

Example:
3x ≤ 9
\(\frac{3x}{3}\hspace{0.5mm}\leq \hspace{0.5mm}\frac{9}{3}\)
x ≤ 3

Multiplication Property of Inequality (when multiplying by a negative number): Multiply both sides of the inequality by the same number.

Example:
\(\frac{x}{-3}\) ≥ 9
\(\frac{x}{-3}\times\hspace{0.5mm}-3 \geq 9\hspace{0.5mm} \times -3\hspace{0.5mm}\)
x ≤ -27

Multiplying or dividing both sides of the inequality by a negative number reverses the sign of the inequality.

In these basic examples, the solution may be evident, but these properties demonstrate how to use an opposite operation to solve for a variable. Using these properties, there are three steps in solving a complex linear inequality.

STEP BY STEP

Step 1. Simplify each side of the inequality. This includes removing parentheses, removing fractions, and adding like terms.

Step 2. Use addition and subtraction properties of an inequality to move the variable to one side of the equation and all number terms to the other side of the equation.

Step 3. Use multiplication and division properties of an inequality to remove the value in front of the variable. Reverse the inequality sign if multiplying or dividing by a negative number.

In Step 2, all of the variables may be placed on the left side or the right side of the inequality. The examples in this lesson will place all of the variables on the left side of the inequality.

Solving Linear Inequalities Review

Let’s Review!

• A linear equation is an equation with one solution. Using opposite operations solves a linear equation.
• The process to solve a linear equation or inequality is to eliminate fractions and parentheses and combine like terms on the same side of the sign. Then, solve the equation or inequality by using inverse operations.