Algebra Know-It-ALL

176 Equations and Inequalities

Are you confused?

To avoid mistakes like the one just described, we must never divide an equation through by any variable

or expression that might happen to equal 0. The safest approach is to divide through only by nonzero

numbers, and never by variables or expressions containing variables.

Here’s a challenge!

Show that we cannot, in general, square both sides of an equation and end up with another equation that

has the same solution.

Solution

To see what can go wrong when we square both sides of an equation, consider this:

x=− 4

This is as simple as an equation can be. It states its own solution, which is −4. If we square both sides, we get

x^2 = 16

This new equation has two solutions, x=−4 and x= 4. It’s clearly not equivalent to the original equation.

We’ve gotten ourselves into trouble, just as we did when we inadvertently divided by 0. But this time,

instead of missing a legitimate solution, we’ve generated a false one!

Inequalities

Most of the time in algebra, you’ll work with equations. These are statements that involve

quantities that are the same, or are supposed to be the same. But sometimes you’ll need to

express the fact that quantities differ, or at least the fact that they don’t have to be the same.

Such statements are called inequalities.

Not equal

When you want to indicate that two quantities are never equal, but you don’t want to specify

relative size or the extent to which they’re different, you can use the “not equal to” symbol.

You’ve already seen this in action. It’s an equals sign with a slash through it (≠). Here are some

examples of its use:

• To state that 3 is not equal to 7/2, write 3 ≠ 7/2.


• To state or require that x is never equal to 0, write x≠ 0.


• To state or require that x is never equal to y, writex≠y.


• To state or require that 2x is never equal to x, write 2x≠x.

Strictly larger

When a certain quantity is always larger than (or greater than) some other quantity, the “strictly

larger than” symbol is used. It looks like a letter V rotated a quarter-turn counterclockwise,