Equations with Algebraic Fractions

The fraction bar works like a bracket: in two x minus one, all over five, the whole of two x minus one is divided by five. To undo dividing by five, multiply both sides by five. The left side becomes two x minus one, and the right side becomes fifteen. Add one to both sides: two x equals sixteen. Divide by two: x equals eight. Check: two eights are sixteen, minus one is fifteen, and fifteen divided by five is three.

When there is a fraction on each side, multiply both sides by the lowest common denominator. For x plus three over four equals two x minus one over three, the LCD of four and three is twelve. Twelve divided by four is three, so the left side becomes three lots of x plus three. Twelve divided by three is four, so the right side becomes four lots of two x minus one. Keep those brackets. Expanding gives three x plus nine equals eight x minus four. Subtract three x and add four: thirteen equals five x, so x is two point six. The answer does not have to be a whole number. Check: both sides come out to one point four.

Here two fractions sit on the same side. The LCD of two and five is ten, so multiply every term by ten. x over two times ten is five x. x over five times ten is two x. And the six on the right is a term as well, so it becomes sixty. Now five x minus two x equals sixty, which is three x equals sixty, so x is twenty. Check: twenty over two is ten, twenty over five is four, and ten minus four is six.

This is where most mistakes happen. In x over two minus x minus four over three, the minus sign belongs to the whole numerator x minus four. Multiply every term by six. x over two becomes three x. The second fraction becomes two lots of x minus four, and it is being subtracted, so write minus two, bracket, x minus four. Now expand carefully: negative two times x is negative two x, and negative two times negative four is positive eight. So three x minus two x plus eight equals eighteen, which gives x plus eight equals eighteen, and x is ten. If you write minus eight, you get twenty-six, and the check fails.

A whole number on its own is still a term, so it gets multiplied by the LCD too. In three x minus one over four, plus two, equals x over two, the LCD is four. The first fraction becomes three x minus one, the two becomes eight, and x over two becomes two x. So three x plus seven equals two x. Subtract two x and subtract seven: x equals negative seven. Check in the original: three times negative seven minus one is negative twenty-two, over four is negative five point five, plus two is negative three point five. And negative seven over two is also negative three point five. Quiz and exercises now.