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**Solving Equations A Solution**

A value of the variable that makes the equation a true statement

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Equations Example: x + 2 = 5 TRUE if x = 3 FALSE if x = anything else The Solution is x = 3

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**Special Cases Example: x = x + 1 NEVER TRUE No such number exists**

Called a contradiction

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**Special Cases Example: 2x = x + x ALWAYS TRUE True for any number**

Called an identity

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Equivalent Equations Have the same solution Example: x + 2 = 5 x – 1 = 2 x + 4 = 7 All have solution x = 3

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Addition Principle Adding (or subtracting) the same number to both sides of an equation does not change its solution.

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**Addition Principle Example: 6 + x = 8 3+6 + x = 3+8 9 + x = 11**

Are equivalent equations Both have the same solution

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**Addition Principle Example: 6 + x = 8 -6 -6 x = 2**

x = 2 Equivalent equation that shows the solution

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**Multiplication Principle**

Multiplying (or dividing) same non-zero number to both sides of an equation does not change its solution.

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**Multiplication Principle**

Example: 6x = 12 3 • 6x = 3 • 12 18x = 36 Are equivalent equations Both have the same solution

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**Multiplication Principle**

Example: 6x = 12 6x 6 = 12 6 x = 2 Equivalent equation that shows the solution

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**Multiplication Principle**

Example:

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**Multiplication Principle**

Another Way:

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Using Both Principles Usually best to use Addition Principle first.

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**Using Both Principles Example: 2x – 3 = 7 First add 3: 2x = 10**

Then 2: x = 5

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Notes 3.4 – SOLVING MULTI-STEP INEQUALITIES

Notes 3.4 – SOLVING MULTI-STEP INEQUALITIES

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