Solving by inspection

Solving by inspection is the use of guessing to solve a maths problem.

For example, to solve , to find x, if you think about it long enough, the answer might POP out at you as being x must be 4. You’ll notice you’ve worked this out by thinking “what added to 7 will give me 11”, so you start with 11 fingers (represented by the 1), and count down 7, obtaining 4. Essentially, to work out the answer, you have used .

The opposite of subtract is addition.

The opposite of multiply is divide. For example, , therefore .

The opposite of square is square root. For example, if , therefore .

Solving equations using algebra

Realizing what we’ve learnt by solving by inspection (which is not systematic enough to be used every time when solving for a question, since an answer rarely POPS out at you!), the steps to solve equations with algebra are (applying to the problem ):

1. Apply distributive law to remove brackets, if necessary
2. Add numbers to (or subtract numbers from) both sides of the equation to simplify
   For example, , therefore
3. Group variables (x) on one side and constants on the other
   For example, , therefore
4. Divide both sides through by numbers to solve for x
   For example,  through-dividing by 5, , therefore

Another example where the distributive law is first necessary is when solving , in this example, you need to in-multiply 4 into the , giving you . Then, you can solve it, with the above steps, to give you .