Here, we can see that the independent variable is the number of stickers, so we'll call that \(\displaystyle x\). Each sticker costs $2 to make, so we'll write that as \(\displaystyle 2x\). The cost of the machine is always the same ($22), so we call that a constant.

So far, we have \(\displaystyle 2x+22\) as the cost of making \(\displaystyle x\) number of stickers.

Now, we need to describe the revenue gained by selling \(\displaystyle x\) number of stickers. We know that Carla wants to sell the stickers at $3 each, which we can write as \(\displaystyle 3x\).

"Breaking even" means that your costs equal your income. In other words, it's the point at which Carla's sold exactly enough stickers to make all of her initial investment back. In math, we use an equal sign to describe this. Hence, we end up with the equation:

\(\displaystyle 3x=2x+22\)

To set up this equation we need to put the amount of money she needs on one side of the equation and the amount of money she earns on the other side of the equation. It looks like the following

\(\displaystyle 23=5x+3\)

where \(\displaystyle x\) represents the number of hours Sally works. To get the dollar amount she earns for the hours she works, we mulitply it by 5.  The 3 represents the amount of money she already has saved. These two added together needs to equal 23 dollars, the amount for the doll.

From here we solve for \(\displaystyle x\):

\(\displaystyle 23=5x+3\)

\(\displaystyle 20=5x\)

\(\displaystyle 4=x\)

Let us call the number of cards Annie has \(\displaystyle x\)

Josh has twice as many so we can call the number of cards Josh has \(\displaystyle 2x\)

Brenda has 3 times as many cards as Josh so we can call the number of cards Brenda has \(\displaystyle 3 \times 2x = 6x\)

The total is 90 so we have \(\displaystyle x + 2x + 6x = 90\)

Which we simplify by adding all the x terms: \(\displaystyle 9x=90\)

The salesman will sell either more than 5 cars or less than 5.

Therefore, only one set of the equations will be true for any one month.

The word between the expressions needs to be "or."

The total profit is expressed by 30,000x.

The commision can be calculated as \(\displaystyle \% \text{commision} \times 30,000x\).

The higher commision must be matched to the equation for greater amount of cars sold, \(\displaystyle x>5\). 

To make this much easier, translate the word problem into a system of three equations.

\(\displaystyle C=3\)

\(\displaystyle S=2C+2\)

\(\displaystyle N=\frac{1}{4}S-1\)

We have C for Chai, S for Sora, and N for Newton. To find Sora's age, plug in \(\displaystyle C=3\) into \(\displaystyle S=2C+2\).

\(\displaystyle S=2(3)+2=8\)

Sora is 8 years old. Use this to find Newton's age.

\(\displaystyle N=\frac{1}{4}(8)-1=1\)

Newton is one year old.  So the answer is:

Sora, 8 years

Newton, 1 year

Let us call x the smallest integer. Because the next two numbers are consecutive even integers, we can call represent them as x + 2 and x + 4. We are told the sum of x, x+2, and x+4 is equal to 72. 

x + (x + 2) + (x + 4) = 72

3x + 6 = 72

3x = 66

x = 22.

This means that the integers are 22, 24, and 26. The question asks us for the product of these numbers, which is 22(24)(26) = 13728. 

The answer is 13728.

Take every word and translate into math. 

\(\displaystyle 2\) more than means that you need to add \(\displaystyle 2\) to something.

That something is \(\displaystyle x\) so just combine them to have an expression of \(\displaystyle x+2\). 

Take every word and translate into math. 

\(\displaystyle 3\) less than means that you need to subtract \(\displaystyle 3\) from something.

That something is \(\displaystyle x\) so just combine them to have an expression of \(\displaystyle x-3\). 

Take every word and translate into math. 

\(\displaystyle 3\) times something means that you need to multiply \(\displaystyle 3\) to something.

That something is \(\displaystyle x\) so just combine them to have an expression of \(\displaystyle 3x.\)

Take every word and translate it into math.

Anytime you take a quotient of \(\displaystyle a\) and \(\displaystyle b\), \(\displaystyle a\) is the in the numerator and \(\displaystyle b\) is in the denominator.

Therefore expression is \(\displaystyle \frac{x}{10}\)