Question 1

Solve the formula for the specified variable.
-$A = \frac { 1 } { 2 } b h \quad \text { for } b$

Accepted Answer

To solve for $b$, multiply both sides by $2$ and divide by $h$, resulting in $b = \frac { 2 A } { h }$.

Question 2

Find all values of x satisfying the given conditions.
-$y _ { 1 } = x , y _ { 2 } = 9 + x , y _ { 3 } = 3 ( x - 5 ) + 10 x$, and the sum of 8 times $y _ { 1 }$ and 4 times $y _ { 2 }$ equals y3.

Accepted Answer

The answer of Find all values of x satisfying the...

Question 3

Find all values of x satisfying the given conditions.
-$y _ { 1 } = \frac { 1 } { x + 4 } , y _ { 2 } = \frac { 1 } { x - 4 } , y _ { 3 } = \frac { 1 } { x ^ { 2 } - 16 }$, and the difference between 4 times $y _ { 1 }$ and 2 times $y _ { 2 }$ is the product of 6 and $y 3$.

Accepted Answer

The given condition can be translated into the equation: $4y_1 - 2y_2 = 6y_3$. Substituting the given expressions for $y_1$, $y_2$, and $y_3$ yields: $4\left(\frac{1}{x+4}\right) - 2\left(\frac{1}{x-4}\right) = 6\left(\frac{1}{x^2-16}\right)$. Simplifying and solving for $x$ gives $x = 15$.

Question 4

Solve the problem.
-The length of a rectangular room is 6 feet longer than twice the width. If the room's perimeter is 216 feet, what are the room's dimensions?

Accepted Answer

The answer of Solve the problem.
-The length of a rectangular...

Question 5

Solve the problem.
-An auto repair shop charged a customer $496 to repair a car. The bill listed $96 for parts and the remainder for labor. If the cost of labor is $40 per hour, how many hours of labor did it take to repair the car?

Accepted Answer

The total cost of the repair was $496, with $96 for parts. Subtracting the cost of parts from the total gives $400 for labor. At $40 per hour, it took $400 / $40 = 10 hours of labor.

Question 6

Solve the problem.
-The sum of the angles of a triangle is 180°. Find the three angles of the triangle if one angle is three times the smallest angle and the third angle is 30° greater than the smallest angle.

Accepted Answer

The answer of Solve the problem.
-The sum of the angles...

Question 7

Solve the problem.
-The president of a certain university makes three times as much money as one of the department heads. If the total of their salaries is $270,000, find each worker's salary.

Accepted Answer

Let the department head's salary be x. The president's salary is 3x. Together, their salaries add up to $270,000, so x + 3x = 4x = $270,000. Solving for x gives x = $67,500 (department head's salary). Therefore, the president's salary is 3x = $202,500.

Question 8

Solve the problem.
-A car rental agency charges $225 per week plus $0.20 per mile to rent a car. How many miles can you travel in one week for $245?

Accepted Answer

The weekly charge is $225, leaving $20 for mileage ($245 - $225). At $0.20 per mile, you can travel 100 miles ($20 / $0.20 per mile).

Question 9

Solve the problem.
-During a road trip, Tony drove one-third the distance that Lana drove. Mark drives 21 more miles than Lana drove. The total distance they drove on the trip was 469 miles. How many miles did each person drive?

Accepted Answer

Let Lana's distance be x. Then Tony's distance is x/3, and Mark's distance is x + 21. The equation is x/3 + x + x + 21 = 469. Solving gives x = 192, so Tony drove 64 miles, Lana drove 192 miles, and Mark drove 213 miles.

Question 10

Find all values of x satisfying the given conditions.
-$y _ { 1 } = \frac { 1 } { x + 4 } , y _ { 2 } = \frac { 1 } { x + 3 } , y _ { 3 } = \frac { - 1 } { x ^ { 2 } + 7 x + 12 }$, and the sum of $y _ { 1 }$ and 2 times $y _ { 2 }$ is $y _ { 3 }$.

Accepted Answer

Given that $y_1 + 2y_2 = y_3$, substituting the given expressions for $y_1$, $y_2$, and $y_3$ yields $\frac{1}{x+4} + \frac{2}{x+3} = \frac{-1}{x^2 + 7x + 12}$. Simplifying the right side of the equation, $x^2 + 7x + 12$ factors into $(x+4)(x+3)$, which are the denominators of the left side. However, the sign on the right side is negative, which cannot be achieved by adding positive fractions from the left side. Therefore, there is no solution.

Question 11

Solve the problem.
-There are 18 more sophomores than juniors in an 8 AM algebra class. If there are 52 students in this class, find the number of sophomores and the number of juniors in the class.

Accepted Answer

Let the number of juniors be $j$ and the number of sophomores be $j + 18$. The total number of students is $j + (j + 18) = 52$. Solving for $j$ gives $2j + 18 = 52$, so $2j = 34$, and $j = 17$. Therefore, there are 17 juniors and $17 + 18 = 35$ sophomores.

Question 12

Solve the problem.
-You inherit $10,000 with the stipulation that for the first year the money must be invested in two stocks paying 6% and 11% annual interest, respectively. How much should be invested at each rate if the total interest earned For the year is to be $700?

Accepted Answer

The answer of Solve the problem.
-You inherit $10,000 with the...

Question 13

Solve the problem.
-Sergio's internet provider charges its customers $8 per month plus 5¢ per minute of on-line usage. Sergio received a bill from the provider covering a 2-month period and was charged a total of $46.50. How many Minutes did he spend on-line during that period? (Round to the nearest whole minute, if necessary.)

Accepted Answer

The total fixed charge for 2 months is $16 ($8 per month). The remaining $30.50 ($46.50 - $16) is the charge for the minutes used. At 5¢ per minute, the number of minutes is $30.50 / $0.05 = 610 minutes.

Question 14

Solve the problem.
-After a 9% price reduction, a boat sold for $29,120. What was the boat's price before the reduction? (Round to the nearest cent, if necessary.)

Accepted Answer

To find the original price before the reduction, let the original price be X. After a 9% reduction, the price is 91% of X. So, 0.91X = $29,120. Solving for X gives X = $29,120 / 0.91 = $32,000.

Question 15

Solve the problem.
-In a recent International Gymnastics competition, the U.S., China, and Romania were the big winners. If the total number of medals won by each team are three consecutive integers whose sum is 33 and the U.S. won More than China who won more than Romania, how many medals did each team win?

Accepted Answer

The total number of medals won by each team are three consecutive integers whose sum is 33. Let x be the smallest integer. Then the next two integers are x + 1 and x + 2. The sum of these three integers is x + (x + 1) + (x + 2) = 3x + 3 = 33. Solving for x, we get x = 10. Therefore, the three consecutive integers are 10, 11, and 12. Since the U.S. won more than China who won more than Romania, the U.S. won 12 medals, China won 11 medals, and Romania won 10 medals.

Question 16

Solve the problem.
-A train ticket in a certain city is $1.50. People who use the train also have the option of purchasing a frequent rider pass for $15.75 each month. With the pass, each ticket costs only $0.75. Determine the number of times in A month the train must be used so that the total monthly cost without the pass is the same as the total monthly Cost with the pass.

Accepted Answer

Let $x$ be the number of times the train is used in a month. Without the pass, the cost is $1.50x$. With the pass, the cost is $15.75 + 0.75x$. Setting these equal to find when the costs are the same gives $1.50x = 15.75 + 0.75x$. Solving for $x$ gives $x = 21$.

Question 17

Solve the problem.
-Andrea is having her yard landscaped. She obtained an estimate from two landscaping companies. Company A gave an estimate of $230 for materials and equipment rental plus $65 per hour for labor. Company B gave an Estimate of $335 for materials and equipment rental plus $50 per hour for labor. Determine how many hours of Labor will be required for the two companies to cost the same.

Accepted Answer

To find the number of hours where both companies cost the same, set their cost equations equal to each other: 230 + 65h = 335 + 50h. Solving for h gives h = 7 hours.

Question 18

Solve the problem.
-You inherit $56,000 from a very wealthy grandparent, with the stipulation that for the first year, the money must be invested in two stocks paying 4% and 10% annual interest, respectively. How much should be invested At each rate if the total interest earned for the year is to be $3200?

Accepted Answer

The answer of Solve the problem.
-You inherit $56,000 from a...

Question 19

Solve the problem.
-A bookcase is to be constructed as shown in the figure below. The height of the bookcase is 3 feet longer than the length of a shelf. If 18 feet of lumber is available for the entire unit (including the shelves, but NOT the back of the bookcase), find the length and height of the unit.

Accepted Answer

The answer of Solve the problem.
-A bookcase is to be...

Question 20

Solve the problem.
-Inclusive of a 6.8% sales tax, a diamond ring sold for $2029.20. Find the price of the ring before the tax was added. (Round to the nearest cent, if necessary.)

Accepted Answer

The answer of Solve the problem.
-Inclusive of a 6.8% sales...

Quiz 1: Equations and Inequalities

Solve the formula for the specified variable. - $A = \frac { 1 } { 2 } b h \quad \text { for } b$

Find all values of x satisfying the given conditions. - $y _ { 1 } = x , y _ { 2 } = 9 + x , y _ { 3 } = 3 ( x - 5 ) + 10 x$ , and the sum of 8 times $y _ { 1 }$ and 4 times $y _ { 2 }$ equals y3.

Find all values of x satisfying the given conditions. - $y _ { 1 } = \frac { 1 } { x + 4 } , y _ { 2 } = \frac { 1 } { x - 4 } , y _ { 3 } = \frac { 1 } { x ^ { 2 } - 16 }$ , and the difference between 4 times $y _ { 1 }$ and 2 times $y _ { 2 }$ is the product of 6 and $y 3$ .

Solve the problem. -The length of a rectangular room is 6 feet longer than twice the width. If the room's perimeter is 216 feet, what are the room's dimensions?

Solve the problem. -An auto repair shop charged a customer $496 to repair a car. The bill listed $96 for parts and the remainder for labor. If the cost of labor is $40 per hour, how many hours of labor did it take to repair the car?

Solve the problem. -The sum of the angles of a triangle is 180°. Find the three angles of the triangle if one angle is three times the smallest angle and the third angle is 30° greater than the smallest angle.

Solve the problem. -The president of a certain university makes three times as much money as one of the department heads. If the total of their salaries is $270,000, find each worker's salary.

Solve the problem. -A car rental agency charges $225 per week plus $0.20 per mile to rent a car. How many miles can you travel in one week for $245?

Solve the problem. -During a road trip, Tony drove one-third the distance that Lana drove. Mark drives 21 more miles than Lana drove. The total distance they drove on the trip was 469 miles. How many miles did each person drive?

Find all values of x satisfying the given conditions. - $y _ { 1 } = \frac { 1 } { x + 4 } , y _ { 2 } = \frac { 1 } { x + 3 } , y _ { 3 } = \frac { - 1 } { x ^ { 2 } + 7 x + 12 }$ , and the sum of $y _ { 1 }$ and 2 times $y _ { 2 }$ is $y _ { 3 }$ .

Solve the problem. -There are 18 more sophomores than juniors in an 8 AM algebra class. If there are 52 students in this class, find the number of sophomores and the number of juniors in the class.

Solve the problem. -You inherit $10,000 with the stipulation that for the first year the money must be invested in two stocks paying 6% and 11% annual interest, respectively. How much should be invested at each rate if the total interest earned For the year is to be $700?

Solve the problem. -After a 9% price reduction, a boat sold for $29,120. What was the boat's price before the reduction? (Round to the nearest cent, if necessary.)

Solve the problem. -Inclusive of a 6.8% sales tax, a diamond ring sold for $2029.20. Find the price of the ring before the tax was added. (Round to the nearest cent, if necessary.)