Solving equations

Two equations (or inequalities, or compound statements) are equivalent if they have the same solutions. That is, any assignment of values to variables that makes either statement true will also make the other statement true.

For example, since x := 5 is the only assignment that will make

2x − 4 = 3x − 9

true, and it's also (obviously) the only assignment that will make

x = 5

true, these two equations are equivalent. Similarly, the equation

t = 1/t

is equivalent to the compound statement

t = 1 or t = −1,

because in each case the only solutions are t := 1 and t := −1. Also, the inequality

2x − 4 > 3x − 9

is equivalent to the inequality

x < 5,

because in each case, the solutions are given by assigning x to any value less than 5.

The book doesn't introduce a symbol to describe equivalence of statements, but you can use ‘⇔’ if you wish. (This symbol is usually read aloud as ‘if and only if’.) For example, the equivalences above can be symbolised thus:

2x − 4 = 3x − 9 ⇔ x = 5;
t = 1/t ⇔ t = 1 or t = −1;
2x − 4 > 3x − 9 ⇔ x < 5.

Now, you don't really need this symbol; as you solve an equation, you usually just list equivalent equations in a column, from your original equation to the final answer. So as far as I'm concerned, you don't have to learn this symbol. However, I do want to make the point that you should not use an equals sign! For example, this would be quite wrong:

2x − 4 = 3x − 9 = x = 5. (WRONG!)

In this problem, you do not want to say that 3x − 9 is equal to 5.

Equivalent equations

The easiest way to turn an equation into an equivalent equation is to replace one side (or both) with an equivalent expression. For example, the expression 2(x + y) is equivalent to the expression 2x + y, so the equation

2(x + y) = 7x

is equivalent to the equation

2x + 2y = 7x

(and of course there's nothing special about the 7x here).

If an expression in an equation has a side condition (like x ≠ 0), then you can remove that condition from the equation if you make a compound statement with ‘and’ that includes the side condition. For example, consider the equation

x³/x = x.

Since x³/x is equivalent to x² for x ≠ 0, the original equation is equivalent to the compound statement

x² = x and x ≠ 0.

Notice that x := 0 is a solution to x² = x, but it is not a solution to this compound statement! (In fact, the only solution to that statement, hence the only solution to the original equation, is x := 1.)

The tricks above have nothing to do with equations as such, and they work just as well for inequalities (or any other statement one might make about real numbers). The big idea for solving equations specifically is this: Do the same invertible operation to both sides. An invertible operation, is any operation on a real number that is always defined and that has another (inverse) operation that will always turn the result back into the number you started with.

For example, you can add 5 to any number, and you get back where you started if you then subtract 5, so adding 5 is an invertible operation. Or you can add x to any number, and you get back where you started if you then subtract x, so adding x is an invertible operation. In fact, you can add any defined expression you like, and you get back where you started if you then subtract that same expression, so adding any defined expression is always an invertible operation.

Other invertible operations include subtracting any expression, multiplying by any expression known to be nonzero, and dividing by any expression known to be nonzero. For multiplication and division, it's important that you know that the expression is nonzero; usually, this means that you can only multiply or divide by constants. In general, you can't divide by x, because if x is 0, then this operation is not defined. And you can't multiply by x, because if x is 0, then this operation is not invertible. (However, sometimes you're in a situation where you know that x can't be 0; then it's OK to multiply or divide by x.)

Finally, you can always turn an equation into an equivalent equation by swapping the two sides. This may seem silly, but it's sometimes nice.

In summary, here's a list of techniques for solving equations:

Replace either side (or both) with an equivalent expression;
Add the same expression to both sides (and then simplify them);
Subtract the same expression from both sides (and simplify);
Multiply both sides by the same nonzero expression (and simplify);
Divide both sides by the same nonzero expression (and simplify);
Swap the sides.

(In this list, I've added what is almost always the next step after doing an invertible operation: simplify each side by replacing it with an equivalent expression.)

I'll discuss later another technique useful when you have an absolute value in an expression. You'll learn some others in Intermediate Algebra. There are always more techniques, some of which are still being discovered.

Equivalent inequalities

To solve a pure inequality, that is an inequality with ‘≠’ stating only that two expressions are unequal, simply use the same techniques as for solving an equation.

To solve an order inequality, in contrast, is a little trickier. The first technique —replacing one expression by an equivalent expression— is exactly the same. The last technique —swapping the sides— is almost the same, but you also have to reverse the direction on the inequality. (For example, 2 < x is equivalent to x > 2.) But the main technique —doing the same invertible operation to each side— is more complicated; the operation must also be preserve order.

An order-preserving operation is any operation on a real number that always takes smaller numbers to smaller numbers and larger numbers to larger numbers. Notice that this is relative; adding a million may take small numbers to large numbers, but as long as one number is smaller than another, then it will still be smaller after you add a million to both of them.

Addition and subtraction always preserve order (as long as you are subtracting from the two sides of the inequality). Multiplication or division by a positive quantity is also order-preserving. But multiplication or division by a negative quantity does not preserve order! For example, start with 2 and 3; notice that 2 < 3. After you multiply these by the negative number −4, you get (−4)2 = −8 and (−4)3 = −12, but −8 > −12.

In fact, multiplication or division by a negative quantity reverses order. An order-reversing operation is any operation on a real number that always takes smaller numbers to larger numbers and larger numbers to smaller numbers. You can still use such an operation to solve order inequalities, but you must reverse the direction of the inequality when you do so!

So here's a summary of techniques for solving order inequalities:

Replace either side (or both) with an equivalent expression;
Add the same expression to both sides (and then simplify them);
Subtract the same expression from both sides (and simplify);
Multiply both sides by the same positive expression (and simplify);
Divide both sides by the same positive expression (and simplify);
Multiply both sides by the same negative expression and reverse the inequality;
Divide both sides by the same negative expression and reverse the inequality;
Swap the sides and reverse the inequality.

Compound statements

To solve compound statements, you can just treat each statement separately. But they can also interact if it helps you. For example, go back to the equation x³/x = x. Earlier I changed that to

x² = x and x ≠ 0.

I'd like to divide both sides of the first equation by x, but I can't do that in case x might be zero. But since the inequality x ≠ 0 must also be true, I know that x really is not zero, so in this particular case it's OK to divide by x. If I do that (and simplify), then I get

x = 1 and x ≠ 0,

which obviously equivalent to just x = 1.

In summary:

x³/x = x;
x² = x and x ≠ 0;
x = 1 and x ≠ 0;
x = 1.

If you have a compound inequality, like

2 < x + 1 < 5,

then instead of doing the same order-preserving operation to both sides, you simply do that operation to all sides. So if I subtract 1 from all sides, then I get

1 < x < 4,

so that's the answer.

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