To find the range of x – 2y, you first need to find the range of –2y. You can find the range of –y by multiplying the range of y by –1 (remember to switch the greater than signs to less than signs): –2 < –y < 5. The range of –2y is the range of –y multiplied by 2: –4 < –2y < 10. Adding the range of –2y to the range of x yields the answer:
You can solve this problem by either plugging in or by using algebraic technique. Plugging in the answer choices will get you an answer but will probably take more time than working out the algebra (provided you have a good understanding of the algebra). To work out this problem algebraically, divide the equation into two parts to solve for x. First, let (5x + 2) be positive:
Next, let (5x + 2) be negative:
The range of possible values of x has
two parts: –∞ < x < –1 or
To answer this question quickly, use the method of simultaneous equations. Subtract the second equation from the first:
This question tests rates but also involves percent changes. To find the total distance Tony runs you must find his two rates for each of the two hours of his run. First, we are told that Tony runs for one hour at a rate of 8 miles per hour. His second rate equals the first rate increased by 25 percent, which works out to:
Since Tony runs for one hour at each rate, the distance he runs is equal to:
Since the price of the book does not drop by the same percent every month, you cannot solve this problem as an exponential decay problem. Instead, calculate the percent decreases one by one. After one month, the book’s price is
After two months, its price is
After three months, its price is
Be careful that you don’t try to solve this problem using the exponential decay model. Every hour a constant amount of water drains, not a constant percentage of the remaining water. So the first step in answering the question is determining how much water is left after 19 hours of draining.
During the 20th hour of draining, 800 of the remaining 4800 gallons drain from the tank.
This problem tests your ability to FOIL complicated binomials. The second binomial is complicated because the first term (3y – 2) is itself a binomial. Even so, you can use that term as a unit and distribute according to FOIL in the usual way:
The question asks you to solve a quadratic equation, but the equation cannot be easily factored. Use the quadratic equation to solve for x:
The question asks you to simplify the fraction. To do so, factor the numerator and the denominator to see if they have any terms in common.
The (x + 2) terms on the top and bottom
cancel each other out, and you’re left with the fraction
According to the binomial theorem, for the binomial (a + b)n there are n + 1 terms. So, for this question, there are 8 + 1 = 9 terms in the expansion. Also from the theorem, we know that the power to which a is raised decreases by one each term, beginning with n and ending with 0, so the exponent of a is 9 – 5 = 4. Since the sum of the exponents for each term of the binomial is n, the exponent of b is 8 – 4 = 4. The coefficient of the term is the fifth number in the ninth row of Pascal’s triangle, which is 70. Therefore, the term is 70a4b4.