Solve the equation (2x + 3 / x - 4) - (2x - 8 / 2x + 1) = 1

     Solve the equation  (2x + 3 / x - 4) - (2x - 8 / 2x + 1) = 1 

Answer

To start off, we must find the common denominator between the two fractions.

When dealing with complex problems, which include letters and numbers, it is easiest to multiply each fraction's denominator together like this:

(x - 4) (2x + 1)

This will be the common denominator of the left hand side of the equal sign.

In order to account for this new denominator, we must change the numerator as well.

For example, we must multiply (2x + 3) by (2x +1) and then (2x - 8) by (x - 4) in order to leave the entire equation over the common denominator of (x - 4)(2x + 1)

So the new equation looks as follows:

(2x + 3)(2x + 1) - (2x - 8)(x - 4) / (x - 4)(2x + 1) = 1

We now have to expand the brackets:

(2x + 3)(2x + 1) will become (4x² + 2x + 6x + 3) as you start by multiplying the first number in the first bracket with the first number in the second bracket, then you multiply the first number in the first bracket with the second number in the second bracket, then you multiply the second number in the first bracket with the first number in the second bracket, then finally you multiply the second number in the first bracket with the second number in the second bracket.

(4x² + 2x + 6x + 3) can then be simplified to (4x² + 8x + 3)

(2x - 8)(x - 4) will become (2x² - 8x - 8x + 32), as you follow the same process as above. This can then be simplified to (2x² - 16x + 32)

The common denominator of (x - 4)(2x + 1) will become (2x² + x - 8x - 4), which can be simplified further to (2x² - 7x - 4)

So now the equation overall looks like this:

(4x² + 8x + 3) - (2x² - 16x + 32) / (2x² - 7x - 4) = 1

Now you remove the brackets and simplify the numerator. NB - it is extremely important when removing the brackets of the second equation that you take into account the '-' sign

So the numerator looks like 4x² + 8x + 3 - 2x² +16x (as the double minus makes a plus) - 32

This is then equal to 2x² + 24x - 29

The equation now looks like this:

2x² + 24x - 29 / 2x² - 7x - 4 = 1

Now you must look to remove the denominator. In order to do this you must move it to the other side of the equal's sign to where the one is. That is done by multiplying the 1 with (2x² - 7x - 4)

So,

2x² + 24x - 29 = 2x² - 7x - 4

In order to rearrange this, the 2x² will cancel out, the - 7x will become a + 7x when moved to the left hand side of the equation and it will become + 25 when the - 29 becomes a + 29 after it joins the right hand side of the equation.

31x = 25

Now finally in order to isolate the x you must divide the x by 31

x = 25/31

Therefore, x = 0.8064516129 or 0.81 (2 decimal places)