Equations and Inequalities
Multiplication and Division
Solve the equation 8x = 5x + 24.
The first thing we do is subtract 5x from each side to find that 3x = 24.
We haven't talked yet about what to do with this sort of equation, but you can figure it out by thinking of the balance scale. If three copies of x weigh a total of 24, then each x must weigh one third of 24. That is, x = 24 ÷ 3 = 8. Hopefully we aren't still talking about hacky sacks, because our ankles probably can't handle that much weight.
Since 3x = 24, x = 8 is the solution to the equation 8x = 5x + 24.
In the previous example, we got to the equation 3x = 24. We wanted an equivalent equation with x all by itself, that is, with a coefficient of 1, on the left-hand side of the equation.
To arrive at such an equation, we multiplied both sides of the equation by . It is no coincidence that this is the multiplicative inverse of 3. Did you have an "aha!" moment? Because we did.
In general, whenever we find ourselves with an equation of the form (coefficient) x = (some value), we multiply both sides of the equation by the reciprocal of the coefficient. Since any number times its reciprocal equals 1 (just try finding a number where this doesn't work—we double-dog dare you), this fact leaves us with an equation that has x all by itself on one side, and the solution to the equation on the other side.
As with addition and subtraction, multiplying or dividing both sides of an equation by the same quantity is okay, so long as we are not dividing by zero. Remember: it is all about doing to one side what you do to the other. If you give the left side a treat, you better have brought enough to share with the whole class.
Be careful: When solving an equation, whatever operation you perform on one side you must also perform on the other side. We know we've already said this, but we're hoping the 47th time's the charm. If you add 5 to the left-hand side of an equation, you must add 5 to the right-hand side of the equation. If you divide the right-hand side of the equation by 3, you must also divide the left-hand side of the equation by 3.
Solve the equation 5x - 10 = 3x + 8.
First, get all the x's on one side of the equation by subtracting 3x from each side:
2x - 10 = 8.
Then add 10 to each side to find that
2x = 18.
Finally, divide each side of the equation by 2 to find that
x = 9.
Recall that dividing by 2 is the same as multiplying by one-half. If you would like to see someone give you the fish-eye, try ordering a footlong from Subway and then asking if they can multiply it by one-half. We can think of dividing both sides of an equation by 2, or we can think of multiplying both sides of an equation by —so long as we do the arithmetic right, either way we will find the same answer in the end. However, when fractions are involved, it is usually better to think of multiplying by a multiplicative inverse than to think of dividing by a fraction.
Solve the equation
We want y all by itself, so multiply both sides of the equation by the multiplicative inverse of , which is , to find that
which means y = 10.
If you would prefer to think of dividing both sides of the equation by , by all means, do so. Be on the lookout for any division snakes that might try to bite your ankles. Warning: these snakes may come disguised as numbers.
One thing that can trip us up if we aren't careful is notation with negative signs. When -x appears in an equation, x is multiplied by -1. If it is helpful for you to re-write -x as (-1)x, do it. If anyone laughs at you for it, give us their name and the situation will be taken care of. We're not saying we'll do something to them, we're just saying...taken care of.