Solve the equation for x

Question

. Enter the answer in simplest form.

3(6x−1)+4x=4(3x−1)−5

Answer 1

To solve the equation \(3(6x - 1) + 4x = 4(3x - 1) - 5\), let's start by simplifying both sides step by step.

Step 1: Distribute on both sides.

On the left-hand side: \[ 3(6x - 1) = 18x - 3 \] So the left-hand side becomes: \[ 18x - 3 + 4x = 22x - 3 \]

On the right-hand side: \[ 4(3x - 1) = 12x - 4 \] So the right-hand side becomes: \[ 12x - 4 - 5 = 12x - 9 \]

Now the equation is: \[ 22x - 3 = 12x - 9 \]

Step 2: Combine like terms.

Subtract \(12x\) from both sides: \[ 22x - 12x - 3 = -9 \] This simplifies to: \[ 10x - 3 = -9 \]

Step 3: Isolate \(x\).

Add 3 to both sides: \[ 10x = -9 + 3 \] This simplifies to: \[ 10x = -6 \]

Now, divide both sides by 10: \[ x = -\frac{6}{10} = -\frac{3}{5} \]

So, the solution in simplest form is: \[ \boxed{-\frac{3}{5}} \]

Answer 2

A pump is used to fill pool toys at a rate of 2,160 cubic inches per second. One beach ball takes 2.5 seconds to fill. To the nearest tenth, what is the radius of the beach ball? Use 3.14 for π
. Enter the answer.

Answer 3

To find the radius of the beach ball, we first need to determine its volume based on the filling rate and the time it takes to fill a beach ball.

Step 1: Calculate the volume of the beach ball.

The rate of filling is \(2,160\) cubic inches per second, and it takes \(2.5\) seconds to fill one beach ball. Thus, the volume \(V\) of the beach ball can be calculated as follows:

\[ V = \text{rate} \times \text{time} = 2,160 , \text{cubic inches/second} \times 2.5 , \text{seconds} \]

Calculating that gives:

\[ V = 2,160 \times 2.5 = 5,400 , \text{cubic inches} \]

Step 2: Use the formula for the volume of a sphere to find the radius.

The volume \(V\) of a sphere is given by the formula:

\[ V = \frac{4}{3} \pi r^3 \]

We can set this equal to the volume we calculated:

\[ 5,400 = \frac{4}{3} \pi r^3 \]

Step 3: Solve for \(r^3\).

Substituting \( \pi \) with \(3.14\):

\[ 5,400 = \frac{4}{3} \cdot 3.14 \cdot r^3 \]

Calculating \( \frac{4}{3} \cdot 3.14 \):

\[ \frac{4 \cdot 3.14}{3} \approx 4.1867 \]

So the equation becomes:

\[ 5,400 = 4.1867 r^3 \]

Now we solve for \(r^3\):

\[ r^3 = \frac{5,400}{4.1867} \]

Calculating that value:

\[ r^3 \approx 1,287.1691 \]

Step 4: Take the cube root of \(r^3\) to find \(r\).

Now we find the cube root:

\[ r \approx \sqrt[3]{1,287.1691} \]

Using a calculator:

\[ r \approx 10.7 \]

Final Answer:

Therefore, the radius of the beach ball, to the nearest tenth, is:

\[ \boxed{10.7} \text{ inches} \]

Answer 4

Manuel is playing a board game where he places two characters on a grid. One is placed at the point (−4, 4) and the other is placed at (8, −7). To the nearest tenth, what is the distance between the two characters? Enter the answer.

Answer 5

To find the distance between the two points \((-4, 4)\) and \((8, -7)\) on a grid, we can use the distance formula:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

In this case:

• \(x_1 = -4\), \(y_1 = 4\)
• \(x_2 = 8\), \(y_2 = -7\)

Now we can substitute these values into the formula:

\[ d = \sqrt{(8 - (-4))^2 + (-7 - 4)^2} \]

Calculating the differences:

\[ d = \sqrt{(8 + 4)^2 + (-7 - 4)^2} \] \[ d = \sqrt{(12)^2 + (-11)^2} \] \[ d = \sqrt{144 + 121} \] \[ d = \sqrt{265} \]

Now, we can approximate \(\sqrt{265}\):

\[ \sqrt{265} \approx 16.2788 \]

Rounding to the nearest tenth:

\[ d \approx 16.3 \]

Thus, the distance between the two characters is:

\[ \boxed{16.3} \]