Lesson 2.4 Creating & Solving Equations & Inequalities in One Variable

Recall the Steps to creating equations & Inequalities Read the problem statement first. Identify the known quantities. Identify the unknown quantity or variable. Create an equation or inequality from the known quantities and variable(s). Solve the problem. Interpret the solution of the equation in terms of the context of the problem.

Example 1 Suppose two brothers who live 55 miles apart decide to have lunch together. To prevent either brother from driving the entire distance, they agree to leave their homes at the same time, drive toward each other, and meet somewhere along the route. The older brother drives cautiously at an average speed of 60 miles per hour. The younger brother drives faster, at an average speed of 70 mph. How long will it take the brothers to meet each other?

Guided Practice: Example 3, continued Read the statement carefully. Identify the known quantities. Identify the unknown variables. Suppose two brothers who live 55 miles apart decide to have lunch together. To prevent either brother from driving the entire distance, they agree to leave their homes at the same time, drive toward each other, and meet somewhere along the route. The older brother drives cautiously at an average speed of 60 miles per hour. The younger brother drives faster, at an average speed of 70 mph. How long will it take the brothers to meet each other?

Create an equation or inequality from the known quantities and variable(s). The distance equation is d = rt or rt = d. Together the brothers will travel a distance, d, of 55 miles. (older brother’s rate)(t) + (younger brother’s rate)(t) = 55 1.2.1: Creating Linear Equations in One Variable

Guided Practice: Example 3, continued The rate r of the older brother = 60 mph and The rate of the younger brother = 70 mph. (older brother’s rate)(t) + (younger brother’s rate)(t) = 55 60t + 70t = 55

Solve the problem for the time it will take for the brothers to meet each other. It will take the brothers 0.42 hours to meet each other. 60t + 70t = 55 Equation 130t = 55 Combine like terms 60t and 70t. Divide both sides by 130. t = 0.42 hours This was a slide, but I think we should just talk about it… Note: The answer was rounded to the nearest hundredth of an hour because any rounding beyond the hundredths place would not make sense. Most people wouldn’t be able to or need to process that much precision. When talking about meeting someone, it is highly unlikely that anyone would report a time that is broken down into decimals, which is why the next step will convert the units.

Example 2 The length of a dance floor to be replaced is 1 foot shorter than twice the width. You measured the width to be 12.25 feet. What is the area and what is the most accurate area you can report? 𝐴=𝑙×𝑤 𝑤=12.25 𝑙=2𝑤−1

𝐴=𝑙×𝑤 𝐴=(2𝑤−1)×12.25

$1000 (3 times a day) + $4,000 (grand prize) no more than $25,000 Example 3 A radio station has no more than $25,000 to give away. They have decided to give away $1,000 three times a day every day until they have at least $4,000 left to award as a grand prize. How many days will the contest run? $1000 (3 times a day) + $4,000 (grand prize) no more than $25,000

3000𝑑+4000≤25000

Example 4 It costs Marcus an access fee for each visit to his gym, plus it costs him $3 in gas for each trip to the gym and back. This month it cost Marcus $108 for 6 trips to his gym. How much is Marcus’s access fee per visit? access fee x # of visits + $3 x # of trips = $108

6𝑥+6 3 =108 6𝑥+18=108

Example 5 Jeff is saving to purchase a new basketball that will cost at least $88. He has already saved $32. At least how much more does he need to save for the basketball? saved $32 at least $88

32+𝑥≥88

cost with discount x p = $29.94 Example 6 Rebecca bought p pairs of socks and received a 20% discount. Each pair of socks cost her $4.99. Her total cost without tax was $29.94. How many pairs of socks did Rebecca buy? cost with discount x p = $29.94

4.99𝑝=29.94