## Equations and Inequations

 An equation is made up of two expressions and an equals sign e.g.  + = 2

To solve an equation, the value or values of the variable must be found that make both sides of the equation have the same value.

There are several types of equations and several methods of solving them.

When solving equations each step should be written on a new line, and the equals signs should be kept directly underneath each other.

### Linear Equations

• These equations usually have one variable and one solution.
• Equations can sometimes be solved by inspection (working out the answer mentally).
This method usually works for simple equations only.
• The aim when solving an equation is to get one of the variables on its own, on one side of the equation.
• To remove a term from one side of an equation, carry out the opposite operation to both sides of the equation.

Remember: Adding and subtracting are opposite operations.

• Multiplying and dividing are opposite operations.

 Examples Answers Solve: x + 7 = 15 x + 7 = 15 x + 7 − 7 = 15 − 7 (subtract 7 from both sides) x = 8 g − 3 = 12 g − 3 = 12 g − 3 + 3 = 12 + 3 (add 3 to both sides) g = 15 3w = 15 3w = 15 3w⁄3 = 15⁄3 (divide both sides by 3) w = 5 k⁄4 = 5 k⁄4 = 5 k⁄4 × 4 = 5 × 4 (multiply both sides by 4) k = 20

The answer should then be checked by substituting it back into the equation to make sure that both sides have the same value. This step can be done mentally.

### Equations with Brackets

Generally, it is better to expand any brackets first.

 Solve 3(x + 7) = 24 3(x + 7) = 24 3x + 21 = 24 (expand brackets) 3x + 21 − 21 = 24 − 21 (subtract 21 from both sides) 3x = 3 (divide both sides by 3) x = 1

### Equations with Fractions

There are two ways to deal with fractions:

1. Multiply both sides by the reciprocal.

2. Multipy every term by the common denominator.

### Equations with Two Variable Terms

If the variable is on both sides of the equation, collect the variable terms on the side that has the most.

For a great little program for practice with solving equations try this link to another site. Click here.

Courtesy David Hellam, teacher and Szymon Rutkowski, student from Kuwait English School (permission pending)

### Special Equations

• Some equations have no solution.

e.g. x + 3 = x + 4

x = { } (the empty set)

• Some equations have an infinite number of solutions.

e.g. 3(x + 4) = 3x + 12

3x + 12 = 3x + 12
x can have any value.

### Problem Solving

Many problems can be solved by converting them to algebraic equations and solving them.

The general way to approach these problems is:

• Assign a variable to the unknown quantity in the problem.
• Make up an equation from the information in the problem.
• Solve the equation.

 Example Answer Hemi thinks of a number, doubles it and subtracts 8. The result is 32. What is his number? Let the unknown number be x. The equation is: 2x − 8 = 32 Solving the equation: Hemi's number is 20

### Inequations

Inequations, or inequalities, can be solved in the same way as equations.

 Solve    4x + 6 ≤ 14 (subtract 6 from both sides) 4x ≤ 8 (divide both sides by 4) x ≤ 2

The only difference between equations and inequations is that when both sides of the inequation are

multiplied or divided by a negative number, the inequality sign must be reversed.

 Solve    -3x ≥ 12 (divide both sides by -3 and reverse the inequality sign) x ≤ 12/-3 x ≤ -4