An equation is a mathematical sentence that contains an equals sign. If an equation contains variables, a value for the variable that makes the equation true is a solution to the equation.   To test if a value is a solution of an equation, substitute the value for the variable.

To solve equations, you have to perform inverse operations to get the variable by itself. Remember, addition and subtraction are inverse operations.   If an equation contains addition, solve it by subtracting from both sides to undo the addition.   Similarly, if an equation contains subtraction, solve it by adding to both sides to undo the subtraction.

Multiplication and division are inverse operations.   If an equation contains multiplication, solve it by dividing both sides to undo the multiplication.   Similarly, if an equation contains division, solve it by multiplying both sides to undo the division.

An inequality is a statement that two quantities are not equal.   Inequalities may contain a variable.   The same properties that are used to solve equations are used to solve inequalities. Values of the variable that make the inequality true are solutions to the inequality.

We will graph the solutions on a number line.

• If the inequality is < or >, you’ll use an open circle to mark the number portion of the inequality.
• If it is ≤ or ≥, you’ll use a closed circle.

When two quantities have a constant (unchanged) ratio, their relationship is known as  direct variation.  Many individuals say that one variable “varies directly” as the other. The constant ratio is called the  constant of variation. A direct variation relationship can be represented by a linear equation in the form y = kx, where k is called the constant of proportionality (or constant of variation).

Practice

1) 28x = 252

2) -x – 4 = 11

3) x + 3 = 100

4) ½ x  = 20

5) −1/3x ≤ 20

6) 4x ≥ -125

7)  Write an inequality to express how you’d find the answer and then solve. Carrie is hosting a birthday in a room that holds 140 people.  She has tables that seat 6 people each.  What is the greatest number of tables she can use in the room?

8)  Write an inequality to express how you’d find the answer and then solve. As part of her fitness routine, Mia wants to consume no more than 1,800 calories per day.  She intends to eat 6 small meals today.  What is the average number of calories she can consume per meal?

9)  An elevator can hold no more than 2,500 pounds.  What is the greatest number of people, averaging 140 pounds each, that can safely get into the elevator?
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