When hired for a new job selling jewelry,
you are given two pay options.
Option A, a base salary of $15,000 a year,
with a commission of 10% of your sales,
or option B, a salary of $18,000 a year,
with a commission of 5% of your sales.
We want to determine in order for option A
to produce a larger income, you would need to sell more
than how many dollars worth of jewelry?
Let's begin by letting x equal the amount
of sales in dollars.
And now I need to write an expression
for the income using option A, as well as option B.
So for option A, there's a base salary of $15,000 a year,
which gives us 15,000.
And then plus a 10% commission of your sales,
which in our case would be 10% of x.
To find 10% of x,
we convert the percent to a decimal and multiply.
10% is equal to 0.1 as a decimal,
and therefore 10% of x is equal to 0.1x,
which means for option A,
the annual salary is 15,000 plus 0.1x.
So this is for option A.
And now let's write an expression for option B.
Option B offers a salary of $18,000 a year,
which gives a constant of 18,000.
And then plus 5% of your sales,
which would be 5% of x.
5% of x is equal
to 0.05 times x or 0.05x.
And therefore the annual income
for option B is 18,000 plus 0.05x.
So this is the expression
for the annual income for option B.
And since we're trying to determine what the sales has to be
so that option A produces a larger income,
we want to know when the expression
15,000 plus 0.1x is greater than
the expression 18,000 plus 0.05x.
So not let's go ahead and solve the inequality.
The first step is to add or subtract
to isolate the variable terms on one side.
Notice right now we have x terms on both sides.
Let's begin by subtracting 0.05x on both sides.
Notice now the x terms are going to be on the left side,
so let's also undo the positive 15,000 here on the left
by subtracting 15,000 on both sides.
And now let's go ahead and simplify.
15,000 minus 15,000 is zero.
0.1x minus 0.05x
is equal to 0.05x,
which is greater than, on the right side,
18,000 minus 15,000 is 3,000.
And 0.05x minus 0.05x is zero.
The last step is to multiply or divide to solve for x.
0.05x means 0.05 times x,
which means for the last step we divide both sides by 0.05.
Simplifying, 0.05 divided by itself simplifies to one.
One times x is x.
X is greater than 3,000 divided by 0.05,
which is equal to 60,000,
which means when the amount of sales is over $60,000,
option A will produce a larger income.
So to complete the sentence,
in order for option A to produce a larger income,
you need to sell more than $60,000
worth of jewelry.
I hope you found this helpful.