Linear equations

A linear equation is an equation whose sides are both linear expressions. Similarly, a linear inequality is an equality shose sides are both linear expressions.

Linear equations and inequalities in one variable can always be solved using this method:

At this point, you should have the answer, with the variable equal to (or less than, whatever) a constant. Failing that, you might have a constant statement (with no variable in it), which will be either true or false; then that is your answer.

Examples

Here is a straightforward example: I'll check this solution too: 5(2) + 2 = 12, and 3(2) + 6 = 12, so this checks. In other words, the solution set for x is {2}.

The next example needs to be simplified first:

Checking, 3(2(3) + 4) = 3(10) = 30, and 2(3) + 2(3 + 9) = 6 + 2(12) = 30, so this checks; the solution set for a is {3}.

Here's an example with some negative numbers:

Checking, 2(−1) −8 = −10, and 5(−1) − 5 = −10, so this checks; the solution set for t is {−1}.

Here's an example where you can skip one step:

To check, 4(3) = 12, and 2(3) + 6 = 12, so this checks; the solution set for x is {3}.

Here's a linear inequality; the basic technique is the same:

To check, 5(−2) + 4 = −6, and 2(−2) − 2 = −6, so the original inequality is just barely false when y := −2 (as it should be). To check the direction of the inequality, I'll try y := −3; then 5(−3) + 4 = −11, and 2(−3) − 2 = −8, and −11 < −8. So this checks. In other words, the solution set for y is (−∞, −2).

Here's an example where I have to change the direction of the inequality:

To check, 2(−2) + 5 = 1, and 3(−2) + 7 = 1, so the original inequality is just barely true when c := −2. To check the direction, try c := −4 this time; then 2(−4) + 5 = −3, and 3(−4) + 7 = −5, and −3 ≥ −5. So this checks; the solution set for c is (−∞, −2].

Degenerate cases

Sometimes, when you subtract a variable term, you'll find that the variable disappears after you simplify. (Actually, it could disappear even earlier, when you first simplify the original expressions.) In this case, there is not much point in continuing with the remaining steps; you will have a statement that is simply either true or false.

Here is an example of a false statement:

So this equation is never true; it has no solutions. In other words, the solution set for n is ∅.

Here is an example of a true statement:

So this equation is always true; every real number is a solution for p. (Actually, you could see this immediately after the expressions were simplified; since then both sides are exactly the same!) In other words, the solution set for p is (−∞, ∞).
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