Solving an equation means figuring out which values of the variable make the equation true. The basic approach to equation solving consists of applying the same operation on both sides of the equation until we come up with another equation that has the variable by itself on one side and an expression not containing that variable on the other.

For example, in the (very simple) equation $$x+3 = 5$$ we subtract $3$ on both sides and obtain the equation $$x=2.$$

The latter equation tells us the solution, and we verify that $x=2$ does indeed solve the original equation by substituting $2$ for $x$ in $x+3 = 5$. Since $2+3$ does equal 5 we have indeed found the solution.

The preceding two paragraphs are deceptively simple, but they describe one of the key ideas of algebra.

The following problems explore a subtlety of the concept of ''applying the same operation on both sides of the equation''. In the above example, the two equations involved are equivalent, i.e., one implies the other. If $x=2$ then $x+3 = 5$. Conversely, if $x+3=5$ then $x$ must be $2$, there is no other solution. Sometimes, however, doing the same thing on both sides of an equation creates a new equation that is not equivalent to the old one.

For example, if $x=3$, then squaring on both sides gives $x^2=9$. It is true that $x=3$ implies that $x^2=9$. But the other direction does not hold, if $x^2=9$ then $x$ may be $-3$ since the square of $-3$ also equals 9. The process of squaring introduces a extraneous (or sometimes called spurious) solution. The existence of such solutions is a major reason why you always check your answer.

In this and the next few problems you are asked to decide whether two equations are equivalent, one implies the other, or neither implies the other. Enter (without the quotation marks) "$==>$" if the left equation implies the right, "$<==$" if the right equation implies the left, "$<==>$" if either equation implies the other, and "$><$" if neither equation implies the other. An equation equation A implies an equation B if B is true for all variables for which A is true.

For example, $$x+3 = 5 \quad <==>\quad x = 2$$ $$x=2 \quad==>\quad x^2 = 4$$ $$x^2+6=10 \quad<==\quad x = 2$$ $$x+3 = 5 \quad><\quad x^2 = 9$$ Use these statements in the following items:

$x+3=5$ $x=2$.

$x=2$ $x^2=4$.

$x^2+6=10$ $x=2$.

$x+3=5$ $x^2=9$.