Outro W2

2.6 Solve a Formula for a Specific Variable

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

3.1 Use a Problem-Solving Strategy                                                                              

• Problem-Solving Strategy
  1. Step 1. Read the problem. Make sure all the words and ideas are understood.
  2. Step 2. Identify what we are looking for.
  3. Step 3. Name what we are looking for. Choose a variable to represent that quantity.
  4. 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.
  5. Step 5. Solve the equation using good algebra techniques.
  6. Step 6. Check the answer in the problem and make sure it makes sense.
  7. Step 7. Answer the question with a complete sentence.
• Consecutive Integers
  Consecutive integers are integers that immediately follow each other.
  
  n         1st integer consecutive integer

  n+1      2nd integer consecutive integer

          n+2      3rd integer consecutive integer... etc
  
  Consecutive even integers are even integers that immediately follow one another.
  
  n         1st integer consecutive integer

  n+1      2nd integer consecutive integer

          n+2      3rd integer consecutive integer... etc
  
  Consecutive odd integers are odd integers that immediately follow one another.
  
  n         1st integer consecutive integer

  n+1      2nd integer consecutive integer

          n+2      3rd integer consecutive integer... etc

3.2 Solve Percent Applications                                                                                       

• Percent Increase To find the percent increase:
  1. Step 1. Find the amount of increase. increase=new amount−original amount 
  2. Step 2. Find the percent increase. Increase is what percent of the original amount?
• Percent Decrease To find the percent decrease:
  1. Step 1. Find the amount of decrease. decrease=original amount−new amount 
  2. 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
  
  I=Prt

  where I =i nterest

  P = principal

  r = rate

  t = time
• Discount
  • amount of discount is discount rate ⋅$·$ original price
  • sale price is original price – discount
• Mark-up
  • amount of mark-up is mark-up rate ⋅$·$ original cost
  • list price is original cost + mark up