Solving Linear Equations and Inequalities

Chapter 2

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2.1 – Combining Like Terms2.2 – The Addition Property of Equality2.3 – The Multiplication Property of Equality2.4 – Solving Linear Equations with a Variable on One Side of the Equation2.5 – Solving Linear Equations with a Variable on Both Sides of the Equation2.6 – Formulas2.7 – Ratios and Proportions2.8 – Inequalities in One Variable

Chapter Sections

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Adding with Same SignsTo add real numbers with the same sign,add their absolute values. The sum has the same sign as the numbers being added.

Example:–12 + (–3) = –9 5 + 29 = 34

The sum of two positive numbers will The sum of two positive numbers will always be positive and the sum of two always be positive and the sum of two negative numbers will always be negative.negative numbers will always be negative.

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Adding with Different Signs

To add real numbers with the different signs, subtract the smaller absolute value from the larger absolute value. The sum has the sign of the number with the larger absolute value. Example:12 + (–3) = 9 –28 + 32 = 4The sum of two numbers with different The sum of two numbers with different signs may be positive or negative. The sign signs may be positive or negative. The sign of the sum will be the same as the sign of of the sum will be the same as the sign of the number with the larger absolute value.the number with the larger absolute value.

§ 2.1

Combining Like Terms

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The parts in an algebraic expression that are added are called the terms of the expression.

Expression

-3x + 8y - 15

6w2 + 12z - 31

Terms

-3x, 8y, -15

6w2, 12z, - 31

Terms

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The numerical part of a term is the numerical coefficient or coefficient.

Coefficient

-3

8

, since means x

631

31x

31

Term

-3x

8y

6w2

3x

Terms

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Like terms are terms that have the same variables with the same exponents.

Like Terms

-3x, 8x, - x

6w2, -12w2, w2 31

Unlike Terms

20x, x2, x3

6xy, 2xyz, w2

Like Terms

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Combining Like Terms

1. Determine which terms are like terms.

2. Add or subtract the coefficients of the like terms.

3. Multiply the number found in step 2 by the common variable(s).

Example: 5a + 7a = 12a

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Distributive Property

For any real numbers a, b, and c,a(b + c) = ab + bc

Example: 3(x + 5) = 3x + 15

(This is not equal to 18x! These are not like terms.)

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Simplifying an Expression

1. Use the distributive property to remove any parentheses.

2. Combine like terms.

Example: Simplify 3(x + y) + 2y

= 3x + 3y + 2y (Distributive Property)= 3x + 5y (Combine Like Terms)

(Remember that 3x + 5y cannot be combined because they are not like terms.)