### Key Concepts

## 3.1 Use a Problem-Solving Strategy

**Problem-Solving Strategy**- 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 all the important information. Then, translate the English sentence into an algebra 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.
**Consecutive Integers**

Consecutive integers are integers that immediately follow each other.

$$\begin{array}{cccc}\hfill n\hfill & & & {1}^{\text{st}}\phantom{\rule{0.2em}{0ex}}\text{integer}\hfill \\ \hfill n+1\hfill & & & {2}^{\text{nd}}\phantom{\rule{0.2em}{0ex}}\text{integer consecutive integer}\hfill \\ \hfill n+2\hfill & & & {3}^{\text{rd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive integer . . . etc.}\hfill \end{array}$$

Consecutive even integers are even integers that immediately follow one another.

$$\begin{array}{cccc}\hfill n\hfill & & & {1}^{\text{st}}\phantom{\rule{0.2em}{0ex}}\text{integer}\hfill \\ \hfill n+2\hfill & & & {2}^{\text{nd}}\phantom{\rule{0.2em}{0ex}}\text{integer consecutive integer}\hfill \\ \hfill n+4\hfill & & & {3}^{\text{rd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive integer . . . etc.}\hfill \end{array}$$

Consecutive odd integers are odd integers that immediately follow one another.

$$\begin{array}{cccc}\hfill n\hfill & & & {1}^{\text{st}}\phantom{\rule{0.2em}{0ex}}\text{integer}\hfill \\ \hfill n+2\hfill & & & {2}^{\text{nd}}\phantom{\rule{0.2em}{0ex}}\text{integer consecutive integer}\hfill \\ \hfill n+4\hfill & & & {3}^{\text{rd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive integer . . . etc.}\hfill \end{array}$$

## 3.2 Solve Percent Applications

**Percent Increase**To find the percent increase:- Step 1. Find the amount of increase. $\text{increase}=\text{new amount}-\text{original}\phantom{\rule{0.2em}{0ex}}\text{amount}$
- Step 2. Find the percent increase. Increase is what percent of the original amount?

**Percent Decrease**To find the percent decrease:- Step 1. Find the amount of decrease. $\text{decrease}=\text{original amount}-\text{new}\phantom{\rule{0.2em}{0ex}}\text{amount}$
- Step 2. Find the percent decrease. Decrease is what percent of the original amount?

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

$$\begin{array}{ccc}\hfill I& =\hfill & Prt\hfill \\ \hfill \text{where}\phantom{\rule{1em}{0ex}}I& =\hfill & \text{interest}\hfill \\ \hfill P& =\hfill & \text{principal}\hfill \\ \hfill r& =\hfill & \text{rate}\hfill \\ \hfill t& =\hfill & \text{time}\hfill \end{array}$$**Discount**- amount of discount is discount rate $\xb7$ original price
- sale price is original price – discount

**Mark-up**- amount of mark-up is mark-up rate $\xb7$ original cost
- list price is original cost + mark up

## 3.3 Solve Mixture 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.

$number\xb7value=total\phantom{\rule{0.2em}{0ex}}value$ where*number*is the number of coins and*value*is the value of each coin;*total value*is the total value of all the coins**Problem-Solving Strategy—Coin Word Problems**- Step 1.
**Read**the problem. Make 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 we are looking for. - Step 3.
**Name**what we 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.

## 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem

**Problem-Solving Strategy for Geometry Applications**- Step 1.
**Read**the problem and make all the words and ideas are understood. Draw the figure and label it with the given information. - Step 2.
**Identify**what we are looking for. - Step 3.
**Name**what we are looking for by choosing a variable to represent it. - 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.
**Triangle Properties For $\text{\u25b3}ABC$**

Angle measures:

- $m\text{\u2220}A+m\text{\u2220}B+m\text{\u2220}C=180$

- $P=a+b+c$

- $A=\frac{1}{2}bh,\phantom{\rule{0.2em}{0ex}}\text{b}=\text{base},\text{h}=\text{height}$

**The Pythagorean Theorem**In any right triangle, ${a}^{2}+{b}^{2}={c}^{2}$ where*c*is the length of the hypotenuse and*a*and*b*are the lengths of the legs.**Properties of Rectangles**- Rectangles have four sides and four right (90°) angles.
- The lengths of opposite sides are equal.
- The perimeter of a rectangle is the sum of twice the length and twice the width: $P=2L+2W.$ The area of a rectangle is the length times the width: $A=LW.$

## 3.5 Solve Uniform Motion Applications

**Distance, Rate, and Time***D*=*rt*where*D*= distance,*r*= rate,*t*= time

**Problem-Solving Strategy—Distance, Rate, and Time Applications**- 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 we are looking for. - Step 3.
**Name**what we 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.

## 3.6 Solve Applications with Linear Inequalities

**Solving inequalities**- Step 1.
**Read**the problem. - 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.**Write a sentence that gives the information to find it. Translate into an inequality. - Step 5.
**Solve**the inequality. - 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.