Assignment 2
Practice being neat and organized. Mathematics is a language. Learn to use it correctly and efficiently. Problem #32 on page 22 wants you to solve the equation
An equation is a conditional statement that has no truth value by itself. If we substitute a value for x, then the resulting statement (which in this case is a statement of equality) is either true or false. For instance, if we substitute x = 0 into the equation, we obtain
This is a perfectly meaningful statement. It is, of course, a false statement. If you can classify a statement as false, then the statement is not meaningless. Now, when we solve an equation, our purpose is to find all real number values that make the conditional statement true. We have yet to solve the equation, but we have discovered that x = 0 makes the conditional statement false. Hence, x = 0 is not a solution to the equation. It is not among the collection of numbers that make the conditional statement true. To solve the equation above, we will produce a series of equivalent equations. In doing so, we will preserve the "balance" around the symbol = . (Remember, the Arabic word al-jabr can be translated "to balance."). The symbol ==> means "implies." Let's learn this, because mathematics is a language, and communication is important. It is very important to note that the symbols = and ==> mean two entirely different things. Think before you write! Here is an example of a statement that uses both symbols correctly: 3x = 21 ==> x = 7. To translate, if three times a number is 21, this implies that the number is 7. Or we could say, if 3x = 21 is true, then a necessary consequence is that x = 7. Here is a neat homework-type presentation for problem #32 on page 22.
Translation: If -4(3 + x) + 5 = 4(x+3), then this implies that x must be the number -2.375. Finally, note on page 19 there are what the authors call transformations that produce equivalent equations. Basically, these are principles that you should have learned in first-year algebra. Let's make a real effort to think at a sophisticated level. These transformations, if used properly, preserve the balance around the = symbol. When solving an equation, it is important to preserve the balance. When doing homework, write neatly. Communicate! Say what you mean, and mean what you say. Use the language of mathematics properly, effectively, and efficiently.
Problem: Solve (1/3)(x - 5) = (3/7)(x - 7). Solution (with communication): (1/3)(x - 5) = (3/7)(x - 7) ==> 21[(1/3)(x - 5)] = 21[(3/7)(x - 7)] ==> 7(x - 5) = 9(x - 7) ==> 7x - 35 = 9x - 63 ==> 7x + 28 = 9x ==> 2x = 28 ==> x = 14. Check: Substituting x = 14 into the original equation yields 3 = 3. Problem: You have two summer jobs. In the first job, you work 28 hours and earn $7.25 per hour. In the second job, you earn $6.50 an hour and can work as many hours as you want. If your want to earn $255 per week, how many hours must you work at your second job. Solution (with communication): Let x = the number of hours you need to work at the second job to earn $255 per week. Premise ==> 28($7.25) + x($6.50) = $255 ==> x = [$255 - 28($7.25)]/$6.50 ==> x = 8. Conclusion: You will need to work 8 hours a week at the second job if you want to earn a total of $255 per week. Note: This problem is less time-consuming if you save the calculator computations until the final step. This does require knowledge of the order operations. |