Equations and Inequalities

Equation

An equation is a statement of equality between two algebraic expressions

Examples:

(a)

(b)

(c)

(d)

A solution or root to an equation is a value of the variable that makes the equation a true statement

Example: , x = - 6 is a solution

To solve an equation, isolate the variable on one side of the equation and the non-variable terms on the other side. We do this by using the following properties of equality:

Addition Property of Equality

If a = b and c is any real number then a + c = b + c

Subtraction Property of Equality

If a = b and c is any real number then a – c = b – c

Multiplication Property of Equality

If a = b and c is any real number,

Division Property of Equality

If a = b and c is any real number,

Examples:

(a) Solve:

Subtraction Property

Division Property

(b) Solve:

Remove Parentheses

Addition Property

Subtraction Property

Division Property

Multiply both sides of the equation by the LCD (12)

Multiplication Property

Subtraction Property

(d) Solve:

Addition Property

Subtraction Property

Division Property

(e) Solve:

Remove Parentheses

Combine Like Terms

Addition Property

Subtraction Property

Problems

Solve

(1) 4x – 8 = 12 (7)

(2) 3x + 7 = 10 (8)

(3) 5x + 12 = 3x – 6 (9)

(4) 7x – 4 = 2x + 16 (10) 0.75x – 2.6 = 1.25x + 2.4

(5) 5(x – 2) = 3x – 6 (11) 6x + 10 – 4x = 5x – 8x - 5

(6) 5 = 2(x + 4) – 1 (12) 4 + 3(x – 5) = -4(x + 3) – 6

Answers

(1) x = 5 (7) x = -3

(2) x = 1 (8) x = -10

(3) x = -9 (9) x = -1

(4) x = 4 (10) x = -10

(5) x = 2 (11) x = -3

(6) x = -1 (12) x = -1

Word Problems

(1) Read the problem until it is understandable

(2) Determine what quantity you are trying to find

(3) Represent the quantity by a variable

(4) Write the problem as an equation and solve the equation for the unknown variable

Examples:

(a) Five more than twice a number is 31. What is the number?

Let n = the number

(b) 7 times a number decreased by 5 is 7 more than 3 times the number. What

is the number?

Let n = the number

(c) The length of a rectangle is 3 feet less than twice the width. The perimeter is 30 feet. Find the length of and width of the rectangle.

Let n = the width

2n – 3 = the length

Perimeter of a Rectangle = 2l +2w

Problems

(1) Four times a number decreased by 6 is 3 more than the number. What is the number?

(1) A television set costs $300 more than a DVD player. The total cost of the television set and DVD player is $650. What is the cost of the television set?

(2) The perimeter of a rectangle is 70 feet. The length is 5 feet more than the width. What is the length and width of the rectangle?

(3) A garden is surrounded by 88 feet of fence. The length is 1 foot less than twice the width. What are the length and width of the garden?

Answers

(1) 3

(2) $475

(3) width = 15 feet

length = 20 feet

(4) width = 15 feet

length = 29 feet

Inequalities

An inequality is a statement connecting algebraic expressions by an inequality symbol

Symbols of Inequality

< : less than

: less than or equal to

> : greater than

: greater than or equal to

Examples:

(a) 3x + 1 > 10

(b) x + 3 < -2x – 9

(d) – 4x – 1 2x +11

To solve an inequality, isolate the variable on one side of the inequality and the non-variable term on the other side. We do this by using the following properties of inequalities. The properties hold true for: <, , >, .