Lesson: Chapter - 5

Explanations

1.D

To solve this problem, just isolate the variable:

2.D

Solving for a variable in an inequality is similar to solving for a variable in a normal equation, with one big difference: you isolate the variable. But remember that if you multiply or divide an inequality by a negative number, you must reverse the direction of the inequality symbol:

3.E

First, isolate the expression within the absolute value brackets:

3 + |2x - 7 | = x + 2
|2x - 7 |= x - 1

Then divide the equation into two equations. In this first case, the expression within absolute value brackets is positive:

2x - 7 = x - 1
2x = x + 6
x = 6
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Next, let the expression within absolute value brackets be negative:

2x - 7 = - (x - 1)
2x - 7 = 1 - x
3x = 8
x = 8 /3

The two solutions are x = {⁸/₃, 6}.

4.B

First, solve for s in terms of y:

Next, solve for y in terms of x:

Finally, substitute this value for y in the equation with s:

5.A

This is a rate question, and we are given the input and rate in order to find the output. So, we can plug the known values into the rate formula:

input × rate = output

Jim runs h laps per hour for 2 hours, so his total distance traveled is 2h laps, which equals ^2h/₄ miles. Ryan runs .5h laps per hour for 1.5 hours, so in total he runs 0.75h laps. This is equal to ^.75h/₄ miles. The difference between the distance traveled by Jim and the distance traveled by Ryan is ^2h/₄ – ^.75h/₄ = ^1.25h/₄ miles.

6.B

This is a double percent-change problem, and so we perform each percent change one by one. First, Ken bought the shirt at a discount of 30%. The price at which he paid was:

.7 × 10 = 7 dollars

He then sold it for 60% more than he paid:

1.6 × 7 = 11.2 dollars

He sold the shirt for $11.20.

7.C

The toughest part of this rate problem is translating the word problem into an equation. The point at which the trains will collide is the point at which their combined distance traveled is 255 miles. Using this fact and the rates at which the trains travel, we can find out when the collision occurs, in relation to when the trains left their respective stations. Finally, from this newly calculated information, we can find where the collision occurred. Here is the rate formula we’ll be using:

Distance = Rate × Time

Let x represent the number of hours before the trains collide. We then have the equation:

45 x + 60 (x - 1 ) = 255

This equation explains the situation before the collision: that the train going 45 miles per hour traveled for x hours and the train traveling 60 miles per hour traveled for x – 1 hours. Their combined distance traveled is 255. Now solve the equation for x:

45 x + 60 (x - 1 ) = 255
45 x + 60x - 60 = 255
105 x = 315
x = 3

3 hours pass before the trains collide. From this, we know that the collision happened 3 × 45 = 135 miles from the western station, and 2 × 60 = 120 miles from the eastern station. The halfway point between the stations is ²⁵⁵⁵/₂ = 127.5 miles from either station, so it happened 135 – 127.5 = 7.5 miles from the halfway point between the stations.

8.C

This problem fits the classic exponential decay model. So we plug the given information into the formula:

final amount = origional amount × 1 - decay rate ^{number of changes}

Then we solve:

250 × .96 ^x = 100
.96 ^x = .4
log.96 ^x = log.4
x log.96 ^x = log.4
x = log.96/log.4
x = 22.47

Thus, it takes approximately 22.5 days to reach the 100 pound mark, or, as the question asked, 23 full days.

9.E

To multiply polynomials two at a time, just distribute the terms of one polynomial into the other one individually:

10.C

The equation given is in the form of a quadratic equation ax² + bx + c = 0, so you can use either the reverse FOIL or the quadratic formula to solve for the roots. Before doing either of those things, first factor out 3 from the equation:

3( x² + 8 x - 9) = 0

Factoring takes less time than working out the quadratic formula, so check to see if factoring is possible. It is, and you get:

x² + 8 x - 9 = (x - 1)(x + 9)

The solution set for x is {1, –9}.

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