Solve for x .

To solve for \( x \) in the equation

\[
\frac{x}{-2} = 8,
\]

you can start by multiplying both sides of the equation by \(-2\) to isolate \( x \):

\[
x = 8 \times (-2).
\]

Now, calculate the right side:

\[
x = -16.
\]

So the solution is

\[
\boxed{-16}.
\]

Solve for x

−x over 4 − 2 = −10

To solve the equation

\[
-\frac{x}{4} - 2 = -10,
\]

first, let's isolate the term with \( x \). We can do this by adding \( 2 \) to both sides:

\[
-\frac{x}{4} = -10 + 2.
\]

Calculating the right side gives:

\[
-\frac{x}{4} = -8.
\]

Next, multiply both sides by \(-1\) to get rid of the negative sign:

\[
\frac{x}{4} = 8.
\]

Now, to solve for \( x \), multiply both sides by \( 4 \):

\[
x = 8 \times 4.
\]

Calculating this results in:

\[
x = 32.
\]

Thus, the solution is

\[
\boxed{32}.
\]

Solve for x

5 over 3x − 4 = −9

To solve the equation

\[
\frac{5}{3x} - 4 = -9,
\]

first, we'll isolate the term with \( x \). Start by adding \( 4 \) to both sides:

\[
\frac{5}{3x} = -9 + 4.
\]

Calculating the right side gives:

\[
\frac{5}{3x} = -5.
\]

Next, to eliminate the fraction, multiply both sides by \( 3x \):

\[
5 = -5 \cdot 3x.
\]

This simplifies to:

\[
5 = -15x.
\]

Now, to solve for \( x \), divide both sides by \(-15\):

\[
x = \frac{5}{-15}.
\]

This simplifies to:

\[
x = -\frac{1}{3}.
\]

Thus, the solution is

\[
\boxed{-\frac{1}{3}}.
\]

Solve for x

. 

−5(6x + 15) = −45

To solve the equation

\[
-5(6x + 15) = -45,
\]

first, we can start by dividing both sides of the equation by \(-5\) to simplify it:

\[
6x + 15 = \frac{-45}{-5}.
\]

Calculating the right side gives:

\[
6x + 15 = 9.
\]

Next, we can isolate the term with \( x \) by subtracting \( 15 \) from both sides:

\[
6x = 9 - 15.
\]

Calculating the right side gives:

\[
6x = -6.
\]

Now, divide both sides by \( 6 \):

\[
x = \frac{-6}{6}.
\]

This simplifies to:

\[
x = -1.
\]

Thus, the solution is

\[
\boxed{-1}.
\]

Solve for x: −4(2x − 11) + 7 = 19

To solve the equation

\[
-4(2x - 11) + 7 = 19,
\]

we start by distributing \(-4\) on the left side:

\[
-4 \cdot 2x + 4 \cdot 11 + 7 = 19,
\]

which simplifies to:

\[
-8x + 44 + 7 = 19.
\]

Now combine the constant terms \(44\) and \(7\):

\[
-8x + 51 = 19.
\]

Next, isolate the term with \( x \) by subtracting \( 51 \) from both sides:

\[
-8x = 19 - 51.
\]

Calculating the right side gives:

\[
-8x = -32.
\]

Now, divide both sides by \(-8\):

\[
x = \frac{-32}{-8}.
\]

This simplifies to:

\[
x = 4.
\]

Thus, the solution is

\[
\boxed{4}.
\]

Solve for x: 13x − 1 = 19x − 25

.

To solve the equation

\[
13x - 1 = 19x - 25,
\]

we can start by isolating the \( x \) terms. First, move the \( 19x \) term to the left side by subtracting \( 19x \) from both sides:

\[
13x - 19x - 1 = -25.
\]

This simplifies to:

\[
-6x - 1 = -25.
\]

Next, isolate the term with \( x \) by adding \( 1 \) to both sides:

\[
-6x = -25 + 1.
\]

Calculating the right side gives:

\[
-6x = -24.
\]

Now, divide both sides by \(-6\):

\[
x = \frac{-24}{-6}.
\]

This simplifies to:

\[
x = 4.
\]

Thus, the solution is

\[
\boxed{4}.
\]

Solve ax − 9 = 20

A). x = 29 over a

B). x = 11 over a

C). x = −29 over a

D). x = a over 29

To solve the equation

\[
ax - 9 = 20,
\]

we first isolate the term \( ax \). Start by adding \( 9 \) to both sides:

\[
ax = 20 + 9.
\]

This simplifies to:

\[
ax = 29.
\]

Next, divide both sides by \( a \) to solve for \( x \):

\[
x = \frac{29}{a}.
\]

Looking at the options provided, we can see that the correct answer is:

\[
\boxed{\frac{29}{a}} \quad \text{(Option A).}
\]

The distance formula can be written as d = rt, where d = distance traveled, r = the rate of speed, and t = the length of time traveled. Johan travels 35 miles by bicycle in 1.75 hours. What is Johan’s rate of speed in miles per hour? Explain the solution to the real-world problem.

options:

A). Johan travels at a rate of 61.25 miles per hour.

B). Johan travels at a rate of 20 miles per hour.

C). Johan travels at a rate of 2 miles per hour.

D). Johan travels at a rate of 10 miles per hour.

To determine Johan's rate of speed, we can use the distance formula \( d = rt \), where:

- \( d \) is the distance traveled (in miles),
- \( r \) is the rate of speed (in miles per hour),
- \( t \) is the length of time traveled (in hours).

We know the following values from the problem:

- \( d = 35 \) miles
- \( t = 1.75 \) hours

We need to solve for \( r \). Rearranging the distance formula gives us:

\[
r = \frac{d}{t}.
\]

Substituting the known values into this equation:

\[
r = \frac{35 \text{ miles}}{1.75 \text{ hours}}.
\]

Now, we can perform the calculation:

\[
r = 35 \div 1.75.
\]

To simplify this division, it may help to convert \( 1.75 \) into a fraction. We know that \( 1.75 = \frac{7}{4} \). Thus we can rewrite the expression as:

\[
r = 35 \div \frac{7}{4} = 35 \times \frac{4}{7} = \frac{35 \times 4}{7}.
\]

Calculating \( \frac{35 \times 4}{7} \):

\[
35 \div 7 = 5.
\]

Then,

\[
5 \times 4 = 20.
\]

So, \( r = 20 \) miles per hour.

Thus, Johan travels at a rate of

\[
\boxed{20} \text{ miles per hour} \quad (Option B).
\]

**Explanation of Solution:**

Johan's rate of speed is calculated by taking the total distance traveled (35 miles) and dividing it by the time taken to travel that distance (1.75 hours). This gives us the average speed in miles per hour. In practical terms, knowing the speed helps Johan (or anyone else) understand how fast they are biking and make plans accordingly, such as how long it will take to reach a destination.

Raji draws a square with a side length of x + 5 cm. Ava draws a regular hexagon with a side length of x − 1 cm. The two figures have the same perimeter. Write an equation that allows you to solve for x. What is the value of x ?

To solve for \( x \), we need to establish the perimeters of both the square and the hexagon, then set them equal to each other since the two figures have the same perimeter.

1. **Perimeter of the square**: A square has 4 sides. If each side length is \( x + 5 \) cm, the perimeter \( P \) of the square is:

\[
P_{\text{square}} = 4(x + 5).
\]

2. **Perimeter of the hexagon**: A regular hexagon has 6 equal sides. If each side length is \( x - 1 \) cm, the perimeter \( P \) of the hexagon is:

\[
P_{\text{hexagon}} = 6(x - 1).
\]

3. **Set the perimeters equal**:

\[
4(x + 5) = 6(x - 1).
\]

Now we can solve for \( x \).

### Step 1: Expand both sides

Expanding the left side:

\[
4x + 20
\]

Expanding the right side:

\[
6x - 6
\]

So the equation becomes:

\[
4x + 20 = 6x - 6.
\]

### Step 2: Rearrange the equation

Next, we'll move \( 4x \) to the right side by subtracting \( 4x \) from both sides:

\[
20 = 6x - 4x - 6,
\]

which simplifies to:

\[
20 = 2x - 6.
\]

### Step 3: Isolate the variable

Now, isolate \( 2x \) by adding \( 6 \) to both sides:

\[
20 + 6 = 2x,
\]

which simplifies to:

\[
26 = 2x.
\]

### Step 4: Solve for \( x \)

Now divide both sides by \( 2 \):

\[
x = \frac{26}{2} = 13.
\]

### Conclusion

Thus, the value of \( x \) is

\[
\boxed{13}.
\]

Juwan is thinking of a number. Twice the sum of Juwan’s number and 12 is equal to four less than three times Juwan’s number. What number is Juwan thinking of?

Let's denote Juwan's number as \( x \).

According to the problem statement, we can express the situation with the following equation:

\[
2(x + 12) = 3x - 4.
\]

Now we will solve for \( x \).

### Step 1: Expand the left side

Expanding the left side of the equation:

\[
2x + 24 = 3x - 4.
\]

### Step 2: Rearrange the equation

Now we will rearrange the equation to isolate \( x \). First, let's move \( 2x \) to the right side by subtracting \( 2x \) from both sides:

\[
24 = 3x - 2x - 4.
\]

This simplifies to:

\[
24 = x - 4.
\]

### Step 3: Isolate \( x \)

Next, add \( 4 \) to both sides to solve for \( x \):

\[
24 + 4 = x,
\]

which simplifies to:

\[
x = 28.
\]

### Conclusion

Thus, the number Juwan is thinking of is

\[
\boxed{28}.
\]

Which of the following equations has exactly one solution?

A). 5x + 4 = 5x + 3

B). −5x + 3 = −5x + 3

C). −5x + 4 = 5x + 4

D). 5x + 4 = 5x + 4

To determine which of the equations has exactly one solution, we'll analyze each option one by one.

### Option A:
\[
5x + 4 = 5x + 3
\]
Subtract \( 5x \) from both sides:
\[
4 = 3
\]
This is a false statement. Therefore, this equation has **no solutions**.

### Option B:
\[
-5x + 3 = -5x + 3
\]
Subtract \(-5x\) from both sides:
\[
3 = 3
\]
This is a true statement for all values of \( x \). Therefore, this equation has **infinitely many solutions**.

### Option C:
\[
-5x + 4 = 5x + 4
\]
Let's isolate \( x \):
Add \( 5x \) to both sides:
\[
4 = 10x + 4
\]
Subtract \( 4 \) from both sides:
\[
0 = 10x
\]
Now divide by \( 10 \):
\[
x = 0
\]
This equation has **exactly one solution**: \( x = 0 \).

### Option D:
\[
5x + 4 = 5x + 4
\]
Subtract \( 5x \) from both sides:
\[
4 = 4
\]
This is a true statement for all values of \( x \). Therefore, this equation has **infinitely many solutions**.

### Conclusion
The equation that has exactly one solution is:

\[
\boxed{C} \quad (-5x + 4 = 5x + 4).
\]

Solve for x to determine if the equation has one, none, or infinite solutions. 3(2x + 2) − 4 = −2x + 10

A). The equation has one solution: x = 0

B)The equation has one solution: x = 1
.

To solve the equation

\[
3(2x + 2) - 4 = -2x + 10,
\]

let's start by simplifying both sides.

### Step 1: Expand and simplify the left side

First, distribute the \( 3 \):

\[
6x + 6 - 4 = -2x + 10.
\]

Now combine the constants on the left side:

\[
6x + 2 = -2x + 10.
\]

### Step 2: Combine like terms

Next, we will move all terms involving \( x \) to one side and constant terms to the other side. We'll add \( 2x \) to both sides:

\[
6x + 2x + 2 = 10.
\]

This simplifies to:

\[
8x + 2 = 10.
\]

### Step 3: Isolate the \( x \) term

Now subtract \( 2 \) from both sides:

\[
8x = 10 - 2,
\]

which simplifies to:

\[
8x = 8.
\]

### Step 4: Solve for \( x \)

Now divide both sides by \( 8 \):

\[
x = 1.
\]

### Conclusion

Since we found a specific value for \( x \), the equation has exactly one solution. Thus, the answer is:

\[
\boxed{B} \quad \text{The equation has one solution: } x = 1.
\]

Which answer gives all the values of x for which the inequality 3x + 2 ≥ 9 − 4x is true?

To solve the inequality

\[
3x + 2 \geq 9 - 4x,
\]

we will isolate \( x \) step by step.

### Step 1: Combine like terms

First, we will add \( 4x \) to both sides of the inequality:

\[
3x + 4x + 2 \geq 9.
\]

This simplifies to:

\[
7x + 2 \geq 9.
\]

### Step 2: Subtract 2 from both sides

Next, we will subtract \( 2 \) from both sides:

\[
7x \geq 9 - 2.
\]

This simplifies to:

\[
7x \geq 7.
\]

### Step 3: Divide by 7

Now, divide both sides by \( 7 \):

\[
x \geq 1.
\]

### Conclusion

The solution to the inequality \( 3x + 2 \geq 9 - 4x \) is

\[
x \geq 1.
\]

This means all values of \( x \) that are greater than or equal to 1 satisfy the inequality.

In interval notation, this can be expressed as:

\[
[1, \infty).
\]

So the answer is:

**All values of \( x \) such that \( x \geq 1 \).**