### Key Concepts

## 2.1 Use a General Strategy to Solve Linear Equations

**How 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 equation is true.

If it is true, the number is a solution.

If it is not true, the number is not a solution.

**How to Solve Linear Equations Using a General Strategy**- 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 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 to make sure the result is a true statement.

- Step 1.
Simplify each side of the equation as much as possible.
**How to Solve Equations with Fraction or Decimal Coefficients**- Step 1.
Find the least common denominator (LCD) of
*all*the fractions and decimals (in fraction form) in the equation. - Step 2. Multiply both sides of the equation by that LCD. This clears the fractions and decimals.
- Step 3. Solve using the General Strategy for Solving Linear Equations.

- Step 1.
Find the least common denominator (LCD) of

## 2.2 Use a Problem Solving Strategy

**How To Use a Problem Solving Strategy for Word Problems**- Step 1.
**Read**the problem. Make sure all the words and ideas are understood. - Step 2.
**Identify**what you are looking for. - Step 3.
**Name**what you 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 all the important information. Then, translate the English sentence into an algebra equation. - Step 5.
**Solve**the equation using proper algebra techniques. - Step 6.
**Check**the answer in the problem to make sure it makes sense. - Step 7.
**Answer**the question with a complete sentence.

- Step 1.
**How To Find Percent Change**- Step 1.
Find the amount of change

$\text{change}=\text{new amount}-\text{original amount}$ - Step 2.
Find what percent the amount of change is of the original amount.

$\text{change is what percent of the original amount?}$

- Step 1.
Find the amount of change
**Discount**

$\begin{array}{ccc}\hfill \text{amount of discount}& =\hfill & \text{discount rate}\xb7\text{original price}\hfill \\ \\ \hfill \text{sale price}& =\hfill & \text{original amount}-\text{discount}\hfill \end{array}$**Mark-up**

$\begin{array}{ccc}\hfill \text{amount of mark-up}& =\hfill & \text{mark-up rate}\xb7\text{original cost}\hfill \\ \\ \hfill \text{list price}& =\hfill & \text{original cost}\phantom{\rule{0.2em}{0ex}}+\phantom{\rule{0.2em}{0ex}}\text{mark up}\hfill \end{array}$**Simple Interest**

If an amount of money,*P*, called the principal, is invested or borrowed for a period of*t*years at an annual interest rate*r*, the amount of interest,*I*, earned or paid is:

$$\begin{array}{ccccc}& & I\hfill & =\hfill & \text{interest}\hfill \\ I=Prt\hfill & \phantom{\rule{1em}{0ex}}\text{where}\phantom{\rule{1em}{0ex}}\hfill & P\hfill & =\hfill & \text{principal}\hfill \\ & & r\hfill & =\hfill & \text{rate}\hfill \\ & & t\hfill & =\hfill & \text{time}\hfill \end{array}$$

## 2.3 Solve a Formula for a Specific Variable

**How To Solve Geometry Applications**- Step 1.
**Read**the problem and make sure all the words and ideas are understood. - Step 2.
**Identify**what you are looking for. - Step 3.
**Name**what you are looking for by choosing a variable to represent it. Draw the figure and label it with the given information. - Step 4.
**Translate**into an equation by writing the appropriate formula or model 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.
**The Pythagorean Theorem**- In any right triangle, where
*a*and*b*are the lengths of the legs, and*c*is the length of the hypotenuse, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse.

- In any right triangle, where

## 2.4 Solve Mixture and Uniform Motion Applications

**Total Value of Coins**

For the same type of coin, the total value of a number of coins is found by using the model

$number\xb7value=total\phantom{\rule{0.2em}{0ex}}value$*number*is the number of coins*value*is the value of each coin*total value*is the total value of all the coins

**How to solve coin word problems.**- Step 1.
**Read**the problem. Make sure all the words and ideas are understood.

Determine the types of coins involved.

Create a table to organize the information.

Label the columns “type,” “number,” “value,” “total value.”

List the types of coins.

Write in the value of each type of coin.

Write in the total value of all the coins.

- Step 2.
**Identify**what you are looking for. - Step 3.
**Name**what you are looking for. Choose a variable to represent that quantity.

Use variable expressions to represent the number of each type of coin and write them in the table.

Multiply the number times the value to get the total value of each type of coin. - Step 4.
**Translate**into an equation.

It may be helpful to restate the problem in one sentence with all the important information. Then, translate the sentence into an equation.

Write the equation by adding the total values of all the types of coins. - 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.
**How To Solve a Uniform Motion Application**- Step 1.
**Read**the problem. Make sure all the words and ideas are understood.

Draw a diagram to illustrate what it happening.

Create a table to organize the information.

Label the columns rate, time, distance.

List the two scenarios.

Write in the information you know.

- Step 2.
**Identify**what you are looking for. - Step 3.
**Name**what you are looking for. Choose a variable to represent that quantity.

Complete the chart.

Use variable expressions to represent that quantity in each row.

Multiply the rate times the time to get the distance. - Step 4.
**Translate**into an equation.

Restate the problem in one sentence with all the important information.

Then, translate the sentence into an equation. - 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.

## 2.5 Solve Linear Inequalities

**Inequalities, Number Lines, and Interval Notation**

$x>a\phantom{\rule{3em}{0ex}}x\ge a\phantom{\rule{3em}{0ex}}x<a\phantom{\rule{3em}{0ex}}x\le a$

**Linear Inequality**- A
**linear inequality**is an inequality in one variable that can be written in one of the following forms where*a*,*b*, and*c*are real numbers and $a\ne 0:$

$$ax+b<c,\phantom{\rule{3em}{0ex}}ax+b\le c,\phantom{\rule{3em}{0ex}}ax+b>c,\phantom{\rule{3em}{0ex}}ax+b\ge c.$$

- A
**Addition and Subtraction Property of Inequality**- For any numbers
*a*,*b*, and*c*, if $a<b,\phantom{\rule{0.2em}{0ex}}\text{then}$

$$\begin{array}{cccc}a+c<b+c\hfill & & & a-c<b-c\hfill \\ a+c>b+c\hfill & & & a-c>b-c\hfill \end{array}$$ - We can add or subtract the same quantity from both sides of an inequality and still keep the inequality.

- For any numbers
**Multiplication and Division Property of Inequality**- For any numbers
*a*,*b*, and*c*,

$\begin{array}{c}\text{multiply or divide by a}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{positive}}\hfill \\ \\ \\ \phantom{\rule{2em}{0ex}}\text{if}\phantom{\rule{0.2em}{0ex}}a<b\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c>0,\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}ac<bc\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{a}{c}<\frac{b}{c}.\hfill \\ \phantom{\rule{2em}{0ex}}\text{if}\phantom{\rule{0.2em}{0ex}}a>b\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c>0,\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}ac>bc\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{a}{c}>\frac{b}{c}.\hfill \\ \text{multiply or divide by a}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{negative}}\hfill \\ \\ \\ \phantom{\rule{2em}{0ex}}\text{if}\phantom{\rule{0.2em}{0ex}}a<b\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c<0,\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}ac>bc\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{a}{c}>\frac{b}{c}.\hfill \\ \phantom{\rule{2em}{0ex}}\text{if}\phantom{\rule{0.2em}{0ex}}a>b\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c<0,\phantom{\rule{0.2em}{0ex}}\text{then}\phantom{\rule{0.2em}{0ex}}ac<bc\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\frac{a}{c}<\frac{b}{c}.\hfill \end{array}$

- For any numbers
**Phrases that indicate inequalities**

$>$ $\ge $ $<$ $\le $ is greater than

is more than

is larger than

exceedsis greater than or equal to

is at least

is no less than

is the minimumis less than

is smaller than

has fewer than

is lower thanis less than or equal to

is at most

is no more than

is the maximum

## 2.6 Solve Compound Inequalities

**How to solve a compound inequality with “and”**- Step 1. Solve each inequality.
- Step 2.
Graph each solution. Then graph the numbers that make
*both*inequalities true. This graph shows the solution to the compound inequality. - Step 3. Write the solution in interval notation.

**Double Inequality**- A
**double inequality**is a compound inequality such as $a<x<b$. It is equivalent to $a<x$ and $x<b.$

$\begin{array}{cccc}\text{Other forms:}\hfill & & & \begin{array}{ccccccccccccc}a<x<b\hfill & & & \text{is equivalent to}\hfill & & & a<x\hfill & & & \text{and}\hfill & & & x<b\hfill \\ a\le x\le b\hfill & & & \text{is equivalent to}\hfill & & & a\le x\hfill & & & \text{and}\hfill & & & x\le b\hfill \\ a>x>b\hfill & & & \text{is equivalent to}\hfill & & & a>x\hfill & & & \text{and}\hfill & & & x>b\hfill \\ a\ge x\ge b\hfill & & & \text{is equivalent to}\hfill & & & a\ge x\hfill & & & \text{and}\hfill & & & x\ge b\hfill \end{array}\hfill \end{array}$

- A
**How to solve a compound inequality with “or”**- Step 1. Solve each inequality.
- Step 2. Graph each solution. Then graph the numbers that make either inequality true.
- Step 3. Write the solution in interval notation.

## 2.7 Solve Absolute Value Inequalities

**Absolute Value**

The absolute value of a number is its distance from 0 on the number line.

The absolute value of a number*n*is written as $\left|n\right|$ and $\left|n\right|\ge 0$ for all numbers.

Absolute values are always greater than or equal to zero.**Absolute Value Equations**

For any algebraic expression,*u*, and any positive real number,*a*,

$\begin{array}{ccccc}\text{if}\hfill & & & & \left|u\right|=a\hfill \\ \text{then}\hfill & & & & \phantom{\rule{0.3em}{0ex}}u=\text{\u2212}a\phantom{\rule{0.5em}{0ex}}\text{or}\phantom{\rule{0.5em}{0ex}}u=a\hfill \end{array}$

Remember that an absolute value cannot be a negative number.**How to Solve Absolute Value Equations**- Step 1. Isolate the absolute value expression.
- Step 2. Write the equivalent equations.
- Step 3. Solve each equation.
- Step 4. Check each solution.

**Equations with Two Absolute Values**

For any algebraic expressions,*u*and*v*,

$\begin{array}{ccccc}\text{if}\hfill & & & & \left|u\right|=\left|v\right|\hfill \\ \text{then}\hfill & & & & \phantom{\rule{0.3em}{0ex}}u=\text{\u2212}v\phantom{\rule{0.5em}{0ex}}\text{or}\phantom{\rule{0.5em}{0ex}}u=v\hfill \end{array}$**Absolute Value Inequalities with**$<$ or $\le $

For any algebraic expression,*u*, and any positive real number,*a*,

$\begin{array}{ccccccccccc}\text{if}\hfill & & & & & \left|u\right|<a,\hfill & & & & & \text{then}\phantom{\rule{0.5em}{0ex}}\text{\u2212}a<u<a\hfill \\ \text{if}\hfill & & & & & \left|u\right|\le a,\hfill & & & & & \text{then}\phantom{\rule{0.5em}{0ex}}\text{\u2212}a\le u\le a\hfill \end{array}$**How To Solve Absolute Value Inequalities with**$<$ or $\le $- Step 1. Isolate the absolute value expression.
- Step 2.
Write the equivalent compound inequality.

$\begin{array}{ccccccccccc}\hfill \left|u\right|<a\hfill & & & & & \hfill \text{is equivalent to}\hfill & & & & & \hfill \text{\u2212}a<u<a\hfill \\ \hfill \left|u\right|\le a\hfill & & & & & \hfill \text{is equivalent to}\hfill & & & & & \hfill \text{\u2212}a\le u\le a\hfill \end{array}$ - Step 3. Solve the compound inequality.
- Step 4. Graph the solution
- Step 5. Write the solution using interval notation

**Absolute Value Inequalities with**$>$ or $\ge $

For any algebraic expression,*u*, and any positive real number,*a*,

$\begin{array}{ccccccccccc}\text{if}\hfill & & & & & \left|u\right|>a,\hfill & & & & & \text{then}\phantom{\rule{0.2em}{0ex}}u<\text{\u2212}a\phantom{\rule{0.5em}{0ex}}\text{or}\phantom{\rule{0.5em}{0ex}}u>a\hfill \\ \text{if}\hfill & & & & & \left|u\right|\ge a,\hfill & & & & & \text{then}\phantom{\rule{0.2em}{0ex}}u\le \text{\u2212}a\phantom{\rule{0.5em}{0ex}}\text{or}\phantom{\rule{0.5em}{0ex}}u\ge a\hfill \end{array}$**How To Solve Absolute Value Inequalities with**$>$ or $\ge $- Step 1. Isolate the absolute value expression.
- Step 2.
Write the equivalent compound inequality.

$\begin{array}{ccccccccc}\hfill \left|u\right|>a\hfill & & & & \hfill \text{is equivalent to}\hfill & & & & \hfill u<\text{\u2212}a\phantom{\rule{0.5em}{0ex}}\text{or}\phantom{\rule{0.5em}{0ex}}u>a\hfill \\ \hfill \left|u\right|\ge a\hfill & & & & \hfill \text{is equivalent to}\hfill & & & & \hfill u\le \text{\u2212}a\phantom{\rule{0.5em}{0ex}}\text{or}\phantom{\rule{0.5em}{0ex}}u\ge a\hfill \end{array}$ - Step 3. Solve the compound inequality.
- Step 4. Graph the solution
- Step 5. Write the solution using interval notation