### Key Concepts

## 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality

**To Determine Whether a Number is a Solution to an Equation**- Step 1.
**Substitute the number in for the variable in the equation.** - Step 2.
**Simplify the expressions on both sides of the equation.** - Step 3.
**Determine whether the resulting statement is true.**- If it is true, the number is a solution.
- If it is not true, the number is not a solution.

- Step 1.
**Addition Property of Equality**- For any numbers
*a*,*b*, and*c*, if $a=b$, then $a+c=b+c$.

- For any numbers
**Subtraction Property of Equality**- For any numbers
*a*,*b*, and*c*, if $a=b$, then $a-c=b-c$.

- For any numbers
**To Translate a Sentence to an Equation**- Step 1. Locate the “equals” word(s). Translate to an equal sign (=).
- Step 2. Translate the words to the left of the “equals” word(s) into an algebraic expression.
- Step 3. Translate the words to the right of the “equals” word(s) into an algebraic expression.

**To Solve an Application**- Step 1. Read the problem. Make sure all the words and ideas are understood.
- Step 2. Identify what we are looking for.
- Step 3. Name what we are looking for. Choose a variable to represent that quantity.
- Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with the important information.
- Step 5. Solve the equation using good algebra techniques.
- Step 6. Check the answer in the problem and make sure it makes sense.
- Step 7. Answer the question with a complete sentence.

## 2.2 Solve Equations using the Division and Multiplication Properties of Equality

**The Division Property of Equality**—For any numbers*a*,*b*, and*c*, and $c\ne 0$, if $a=b$, then $\frac{a}{c}=\frac{b}{c}$.

When you divide both sides of an equation by any non-zero number, you still have equality.**The Multiplication Property of Equality**—For any numbers*a*,*b*, and*c*, if $a=b$, then $ac=bc$.

If you multiply both sides of an equation by the same number, you still have equality.

## 2.3 Solve Equations with Variables and Constants on Both Sides

**Beginning Strategy for Solving an Equation with Variables and Constants on Both Sides of the Equation**- Step 1. Choose which side will be the “variable” side—the other side will be the “constant” side.
- Step 2. Collect the variable terms to the “variable” side of the equation, using the Addition or Subtraction Property of Equality.
- Step 3. Collect all the constants to the other side of the equation, using the Addition or Subtraction Property of Equality.
- Step 4. Make the coefficient of the variable equal 1, using the Multiplication or Division Property of Equality.
- Step 5. Check the solution by substituting it into the original equation.

## 2.4 Use a General Strategy to Solve Linear Equations

**General Strategy for Solving Linear Equations**- Step 1.
Simplify each side of the equation as much as possible.

Use the Distributive Property to remove any parentheses.

Combine like terms. - Step 2.
Collect all the variable terms on one side of the equation.

Use the Addition or Subtraction Property of Equality. - Step 3.
Collect all the constant terms on the other side of the equation.

Use the Addition or Subtraction Property of Equality. - Step 4.
Make the coefficient of the variable term to equal to 1.

Use the Multiplication or Division Property of Equality.

State the solution to the equation. - Step 5.
Check the solution.

Substitute the solution into the original equation.

- Step 1.
Simplify each side of the equation as much as possible.

## 2.5 Solve Equations with Fractions or Decimals

**Strategy to Solve an Equation with Fraction Coefficients**- Step 1. Find the least common denominator of all the fractions in the equation.
- Step 2. Multiply both sides of the equation by that LCD. This clears the fractions.
- Step 3. Solve using the General Strategy for Solving Linear Equations.

## 2.6 Solve a Formula for a Specific Variable

**To Solve an Application (with a formula)**- Step 1.
**Read**the problem. Make sure all the words and ideas are understood. - Step 2.
**Identify**what we are looking for. - Step 3.
**Name**what we are looking for. Choose a variable to represent that quantity. - Step 4.
**Translate**into an equation. Write the appropriate formula for the situation. Substitute in the given information. - Step 5.
**Solve**the equation using good algebra techniques. - Step 6.
**Check**the answer in the problem and make sure it makes sense. - Step 7.
**Answer**the question with a complete sentence.

- Step 1.
**Distance, Rate and Time**

For an object moving at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula: $d=rt$ where*d*= distance,*r*= rate,*t*= time.**To solve a formula for a specific variable**means to get that variable by itself with a coefficient of 1 on one side of the equation and all other variables and constants on the other side.

## 2.7 Solve Linear Inequalities

**Subtraction Property of Inequality**

For any numbers a, b, and c,

if $a<b$ then $a-c<b-c$ and

if $a>b$ then $a-c>b-c.$**Addition Property of Inequality**

For any numbers a, b, and c,

if $a<b$ then $a+c<b+c$ and

if $a>b$ then $a+c>b+c.$**Division and Multiplication Properties of Inequalit****y**

For any numbers a, b, and c,

if $a<b$ and $c>0$, then $\frac{a}{c}<\frac{b}{c}$ and $ac>bc$.

if $a>b$ and $c>0$, then $\frac{a}{c}>\frac{b}{c}$ and $ac>bc$.

if $a<b$ and $c<0$, then $\frac{a}{c}>\frac{b}{c}$ and $ac>bc$.

if $a>b$ and $c<0$, then $\frac{a}{c}<\frac{b}{c}$ and $ac<bc$.- When we
**divide or multiply**an inequality by a:**positive**number, the inequality stays the**same**.**negative**number, the inequality**reverses**.