Find the solution to the equation
𝑥 minus seven divided by four multiplied by 𝑥 plus one minus two 𝑥 minus seven
divided by three multiplied by 𝑥 plus one is equal to two.
In order to simplify any algebraic
fractions, when we are adding or subtracting, we need to find the lowest common
denominator. In this case, the 𝑥 plus one is in
both denominators. Therefore, it is already
Common. This means that, in this example,
we’re only interested in the four and the three.
If we consider the fractions
one-quarter and one-third, how do we find the lowest common denominator? We need to find the lowest common
multiple of four and three, the lowest number that is in the four and the three
times table. In this case, our answer is 12. The lowest common multiple of four
and three is 12.
In order to find an equivalent
fraction to a quarter, we need to consider what we’ve done to the bottom. Well, in this case, we’ve
multiplied the bottom or denominator by three. Whatever you do to the bottom you
must do to the top. So we need to multiply the
numerator by three. One multiplied by three is equal to
three. Therefore, the fraction one-quarter
can also be written as three twelfths. In the same way, we can multiply
the top and the bottom of the second fraction by four. This gives us four twelfths. One-third is equivalent to four
As the denominators are now the
same, we just need to subtract the numerators to get our final answer. Three minus four is equal to
negative one. Therefore, our answer is negative
We’re now going to use the same
method to find a common denominator with our algebraic fractions. Firstly, we’ll multiply the top and
the bottom of the first fraction by three. We’ll multiply the top and bottom
of the second fraction by four.
Multiplying the bottom of the first
fraction by three gives us 12 multiplied by 𝑥 plus one. In the same way, multiplying the
bottom of the second fraction by four also gives us 12 multiplied by 𝑥 plus
one. We therefore have a common
denominator. We also need to multiply the top of
the first fraction by three and the top of the second fraction by four. As the denominators on the
left-hand side are the same, we can rewrite this as a single fraction.
Our next step is to expand the
brackets on the top. Three multiplied by 𝑥 is three 𝑥
and three multiplied by negative seven is negative 21. Expanding the second bracket gives
us negative eight 𝑥 plus 28. This is because negative four
multiplied by two 𝑥 is negative eight 𝑥 and negative four multiplied by negative
seven is positive 28. Grouping the like terms on the top
gives us negative five 𝑥 plus seven divided by 12 multiplied by 𝑥 plus one is
equal to two.
Our next step is to multiply both
sides by 12 multiplied by 𝑥 plus one. This cancels the denominator on the
left-hand side. And we are left with negative five
𝑥 plus seven is equal to 24 multiplied by 𝑥 plus one. Expanding the brackets on the
right-hand side gives us 24𝑥 plus 24.
We now need to balance this
equation to calculate 𝑥. Adding five 𝑥 to both sides of the
equation gives us seven is equal to 29𝑥 plus 24. Subtracting 24 from both sides
gives us negative 17 is equal to 29𝑥. And finally, dividing both sides by
29 gives us a value of 𝑥 of negative 17 over 29 or negative seventeen
twenty-ninths. We could check this solution by
substituting in our value of 𝑥 into the initial equation.