Question 1

Solve the equation. $\frac { 2 x - 1 } { 4 } + \frac { 3 x + 2 } { 12 } = \frac { 5 } { 8 }$
A) $\left\{ \frac { 17 } { 18 } \right\}$
B) $\left\{ \frac { 18 } { 17 } \right\}$
C) $\left\{ - \frac { 18 } { 17 } \right\}$
D) $\left\{ - \frac { 17 } { 18 } \right\}$
E) $\varnothing$

Accepted Answer

To solve the equation, first find a common denominator for the fractions, which is 12. This gives: $3(2x-1) + (3x+2) = 5\cdot\frac{12}{8}$. Simplifying, $6x - 3 + 3x + 2 = 15$, leading to $9x - 1 = 15$, and then $9x = 16$, so $x = \frac{16}{9} = \frac{17}{18}$.

Question 2

The cross products in a proportion are equal.

Accepted Answer

The statement "the cross products in a proportion are equal" is known as the Cross Product Property of Proportions and is always true. It can be written as: If a/b = c/d, then ad = bc.

Question 3

Solve the equation. $- 2 ( 3 x - 1 ) - 3 ( 2 x + 5 ) = - 5 ( 2 x - 8 )$
A) $\left\{ - \frac { 53 } { 2 } \right\}$
B) $\left\{ \frac { 53 } { 2 } \right\}$
C) $\left\{ - \frac { 49 } { 2 } \right\}$
D) $\left\{ \frac { 49 } { 2 } \right\}$
E) $\varnothing$

Accepted Answer

Distribute and simplify both sides of the equation to find $x = -\frac{53}{2}$.

Question 4

Equivalent equations are equations that have the same solution set.

Accepted Answer

Equivalent equations are equations that have the same solution set, meaning they yield the same values for the variable(s) in the equation.

Question 5

Solve the equation. $- 5 = 7 x + 2$
A) $\{ 1 \}$
B) $\{ 2 \}$
C) $\{ - 1 \}$
D) $\{ - 2 \}$
E) $\varnothing$

Accepted Answer

First, subtract 2 from both sides to get $-7=7x$. Then, divide both sides by 7 to get $x=-1$. Therefore, the solution is only $\{-1\}$, which corresponds to choice C.

Question 6

Desa invested a certain amount of money at 4% interest and $2,500 more than that amount at 7%. Her total yearly interest was $835. How much did she invest at each rate?
A) $7,000 at 4% and $9,500 at 7%
B) $8,500 at 4% and $6,000 at 7%
C) $6,000 at 4% and $8,500 at 7%
D) $5,000 at 4% and $7,500 at 7%
E) $8,000 at 4% and $10,500 at 7%

Accepted Answer

Let x be the amount invested at 4% interest.

Then, the amount invested at 7% interest is x + 2500.

The total interest earned is the sum of the interest earned from the two investments:

0.04x + 0.07(x + 2500) = 835

Simplifying this equation, we get:

0.04x + 0.07x + 175 = 835

0.11x = 660

x = 6000

Therefore, Desa invested $6,000 at 4% interest and $8,500 (which is $6,000 + $2,500) at 7% interest.

Question 7

Solve the equation. $9 ( 2 x + 3 ) = 7 ( 3 x - 2 )$
A) $\left\{ - \frac { 38 } { 3 } \right\}$
B) $\left\{ \frac { 38 } { 3 } \right\}$
C) $\left\{ \frac { 41 } { 3 } \right\}$
D) $\left\{ - \frac { 41 } { 3 } \right\}$
E) $\varnothing$

Accepted Answer

We can start by simplifying the left and right sides of the equation:
$$9(2x+3)=7(3x-2)$$
$$18x+27=21x-14$$
$$18x-21x=-14-27$$
$$-3x=-41$$
$$x=\frac{41}{3}$$
Therefore, the solution is $\left\{\frac{41}{3}\right\}$.

Question 8

The ratio of male students to female students at a certain university is 5 to 7. If there is a total of 15,840 students, find the number of male and the number of female students.
A) 6,700 males and 9,140 females
B) 6,700 males and 9,040 females
C) 6,800 males and 9,040 females
D) 6,600 males and 9,240 females
E) 9,240 males and 6,600 females

Accepted Answer

The answer of The ratio of male students to female...

Question 9

The sum of the present ages of Eric and his father is 59 years. In 11 years, his father will be twice as old as Eric will be at that time. Find their present ages.
A) Eric is 14 and his father is 45 years old
B) Eric is 16 and his father is 43 years old
C) Eric is 15 and his father is 44 years old
D) Eric is 16 and his father is 45 years old
E) Eric is 13 and his father is 46 years old

Accepted Answer

The answer of The sum of the present ages of...

Question 10

An algebraic equation is called a closed sentence.

Accepted Answer

The answer of An algebraic equation is called a closed...

Question 11

One leg of a right triangle is 4 meters shorter than the other leg. If the length of the hypotenuse is 20 meters, find the length of each leg.
A) 12 meters and 15 meters
B) 13 meters and 16 meters
C) 12 meters and 16 meters
D) 12 meters and 20 meters
E) 13 meters and 15 meters

Accepted Answer

The answer of One leg of a right triangle is...

Question 12

Solve the equation. $\frac { 2 } { 3 } x - \frac { 1 } { 2 } + \frac { 1 } { 4 } x = \frac { 5 } { 8 }$
A) $\left\{ \frac { 25 } { 22 } \right\}$
B) $\left\{ - \frac { 25 } { 22 } \right\}$
C) $\left\{ \frac { 27 } { 22 } \right\}$
D) $\left\{ - \frac { 27 } { 22 } \right\}$
E) $\varnothing$

Accepted Answer

The answer of Solve the equation. \(\frac { 2 }...

Question 13

Solve the equation. $\frac { 2 x - 1 } { 3 } - \frac { 3 x + 2 } { 5 } = - 4$
A) $\{ - 51 \}$
B) $\{ - 47 \}$
C) $\{ - 49 \}$
D) $\{ 49 \}$
E) $\varnothing$

Accepted Answer

First, we must obtain a common denominator for the left side of the equation:

$$\frac{2x-1}{3}-\frac{3x+2}{5}=-4$$ 
$$\frac{10(2x-1)}{30}-\frac{6(3x+2)}{30}=-4$$ 
$$\frac{20x-10-18x-12}{30}=-4$$ 
$$\frac{2x-22}{30}=-4$$

Multiply both sides of the equation by 30 to obtain:

$$2x-22=-120$$ 
$$2x=-98$$ 
$$x=-49$$

Therefore, the solution is $x=-49$, which corresponds to choice C.

Question 14

Find three consecutive integers whose sum is $- 57$ .
A) $- 23 , - 22 , - 21$
B) $- 22 , - 21 , - 20$
C) $- 21 , - 20 , - 19$
D) $- 20 , - 19 , - 18$
E) $- 19 , - 18 , - 17$

Accepted Answer

The answer of Find three consecutive integers whose sum is...

Question 15

When an equation contains fractional forms, multiply both sides by the greatest common factor of all of the denominators.

Accepted Answer

When an equation contains fractional forms, you typically multiply both sides by the least common denominator (LCD) of all the denominators, not the greatest common factor (GCF), to eliminate the fractions.

Question 16

Solve the equation. $0.05 x - 5 ( x + 0.5 ) = 1.2$
A) $\left\{ - \frac { 26 } { 33 } \right\}$
B) $\left\{ \frac { 74 } { 99 } \right\}$
C) $\left\{ - \frac { 74 } { 99 } \right\}$
D) $\left\{ - \frac { 76 } { 99 } \right\}$
E) $\varnothing$

Accepted Answer

We will begin by simplifying the left side of the equation:

\begin{align*}
0.05x - 5(x + 0.5) &= 1.2 \
0.05x - 5x - 2.5 &= 1.2 \
-4.95x &= 3.7 \
x &= \frac{3.7}{-4.95} \
x &= -\frac{74}{99}
\end{align*}

Therefore, the only solution is $x=-\frac{74}{99}$ and the answer is choice C.

Question 17

There are 59 students in a certain class. The number of females is 5 less than three times the number of males. Find the number of females and the number of males in the class.
A) 16 males and 43 females
B) 44 males and 17 females
C) 43 males and 16 females
D) 44 males and 16 females
E) 17 males and 44 females

Accepted Answer

Let the number of males be $m$ and the number of females be $f$. According to the problem, $f = 3m - 5$ and $m + f = 59$. Substituting the first equation into the second gives $m + 3m - 5 = 59$, which simplifies to $4m = 64$, so $m = 16$. Then, $f = 3(16) - 5 = 43$. Therefore, there are 16 males and 43 females.

Question 18

Solve the equation. $\frac { x + 3 } { 3 } + 7 = \frac { x - 3 } { 3 }$
A) $\{ 10.5 \}$
B) $\{ - 9.5 \}$
C) $\{ - 10.5 \}$
D) $\{ 11.5 \}$
E) $\varnothing$

Accepted Answer

The equation simplifies to a contradiction, indicating no solution exists.

Question 19

Sean bought 5 lemons and 3 limes for $3.30. At the same prices, Kim bought 4 lemons and 7 limes for $4.25. Find the price per lemon and the price per lime.
A) $0.65 each for lemons and $0.55 each for limes
B) $0.35 each for lemons and $0.45 each for limes
C) $0.55 each for lemons and $0.45 each for limes
D) $0.45 each for lemons and $0.55 each for limes
E) $0.45 each for lemons and $0.35 each for limes

Accepted Answer

The answer of Sean bought 5 lemons and 3 limes...

Question 20

Solve the equation. $8 - 3 x = 7 + 3 x$
A) $\left\{ - \frac { 1 } { 5 } \right\}$
B) $\left\{ \frac { 1 } { 5 } \right\}$
C) $\left\{ \frac { 1 } { 6 } \right\}$
D) $\left\{ \frac { 1 } { 7 } \right\}$
E) $\varnothing$

Accepted Answer

The answer of Solve the equation. \(8 - 3 x...

Quiz 8: A Transition From Elementary Algebra to Intermediate Algebra

Solve the equation. 2x−14+3x+212=58\frac { 2 x - 1 } { 4 } + \frac { 3 x + 2 } { 12 } = \frac { 5 } { 8 }42x−1​+123x+2​=85​

The cross products in a proportion are equal.

Solve the equation. −2(3x−1) −3(2x+5) =−5(2x−8) - 2 ( 3 x - 1 ) - 3 ( 2 x + 5 ) = - 5 ( 2 x - 8 ) −2(3x−1) −3(2x+5) =−5(2x−8)

Equivalent equations are equations that have the same solution set.

Solve the equation. −5=7x+2- 5 = 7 x + 2−5=7x+2

Desa invested a certain amount of money at 4% interest and $2,500 more than that amount at 7%. Her total yearly interest was $835. How much did she invest at each rate?

Solve the equation. 9(2x+3) =7(3x−2) 9 ( 2 x + 3 ) = 7 ( 3 x - 2 ) 9(2x+3) =7(3x−2)

The ratio of male students to female students at a certain university is 5 to 7. If there is a total of 15,840 students, find the number of male and the number of female students.

The sum of the present ages of Eric and his father is 59 years. In 11 years, his father will be twice as old as Eric will be at that time. Find their present ages.

An algebraic equation is called a closed sentence.

One leg of a right triangle is 4 meters shorter than the other leg. If the length of the hypotenuse is 20 meters, find the length of each leg.

Solve the equation. 23x−12+14x=58\frac { 2 } { 3 } x - \frac { 1 } { 2 } + \frac { 1 } { 4 } x = \frac { 5 } { 8 }32​x−21​+41​x=85​

Solve the equation. 2x−13−3x+25=−4\frac { 2 x - 1 } { 3 } - \frac { 3 x + 2 } { 5 } = - 432x−1​−53x+2​=−4

Find three consecutive integers whose sum is −57- 57−57 .

When an equation contains fractional forms, multiply both sides by the greatest common factor of all of the denominators.

Solve the equation. 0.05x−5(x+0.5) =1.20.05 x - 5 ( x + 0.5 ) = 1.20.05x−5(x+0.5) =1.2

There are 59 students in a certain class. The number of females is 5 less than three times the number of males. Find the number of females and the number of males in the class.

Solve the equation. x+33+7=x−33\frac { x + 3 } { 3 } + 7 = \frac { x - 3 } { 3 }3x+3​+7=3x−3​

Sean bought 5 lemons and 3 limes for $3.30. At the same prices, Kim bought 4 lemons and 7 limes for $4.25. Find the price per lemon and the price per lime.

Solve the equation. 8−3x=7+3x8 - 3 x = 7 + 3 x8−3x=7+3x

Solve the equation. $\frac { 2 x - 1 } { 4 } + \frac { 3 x + 2 } { 12 } = \frac { 5 } { 8 }$

Solve the equation. $- 2 ( 3 x - 1 ) - 3 ( 2 x + 5 ) = - 5 ( 2 x - 8 )$

Solve the equation. $- 5 = 7 x + 2$

Solve the equation. $9 ( 2 x + 3 ) = 7 ( 3 x - 2 )$

Solve the equation. $\frac { 2 } { 3 } x - \frac { 1 } { 2 } + \frac { 1 } { 4 } x = \frac { 5 } { 8 }$

Solve the equation. $\frac { 2 x - 1 } { 3 } - \frac { 3 x + 2 } { 5 } = - 4$

Find three consecutive integers whose sum is $- 57$ .

Solve the equation. $0.05 x - 5 ( x + 0.5 ) = 1.2$

Solve the equation. $\frac { x + 3 } { 3 } + 7 = \frac { x - 3 } { 3 }$

Solve the equation. $8 - 3 x = 7 + 3 x$