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Equations – statements with equal signs – are one type of relationship or comparison between things; they say that terms, expressions, or other entities are exactly the same. An inequality is a bit less precise. Algebraic inequalities show relationships between two numbers, a number and an expression, or between two expressions. In other words, you use inequalities for comparisons.

Inequalities in algebra are less than (<), greater than (>), less than or equal to (< ), and greater than or equal to (> ). A linear equation has only one solution, but a linear inequality has an infinite number of solutions. When you write , for example, you can replace x with 6, 5, 4, –3, –100, and so on, including all the fractions that fall between the integers that work in the inequality.

Here are the rules for operating on inequalities (you can replace the < symbol with any of the inequality symbols, and the rule will still hold):

✔ If a < b, then a + c < b + c (adding any number c).

✔ If a < b, then a – c < b – c (subtracting any number c).

✔ If a < b, then (multiplying by any positive number c).

✔ If a < b, then (multiplying by any negative number c).

✔ If a < b, then (dividing by any positive number c).

✔ If a < b, then (dividing by any negative number c).

✔ If , then (reciprocating fractions).

Notice that the direction of the inequality changes only when multiplying or dividing by a negative number or when reciprocating (flipping) fractions.

You must not multiply or divide each side of an inequality by zero. If you do so, you create an incorrect statement. Multiplying each side of 3 < 4 by 0, you get 0 < 0, which is clearly a false statement. You can’t divide each side by 0, because you can never divide anything by 0 – no such number with 0 in the denominator exists.

Solving linear inequalities

To solve a basic linear inequality, you first move all the variable terms to one side of the inequality and the numbers to the other. After you simplify the inequality down to a variable and a number, you can find out what values of the variable will make the inequality into a true statement. For example, to solve 3x + 4 > 11 – 4x, you add 4x to each side and subtract 4 from each side. The inequality sign stays the same because no multiplication or division by negative numbers is involved. Now you have 7x > 7. Dividing each side by 7 also leaves the sense (direction of the inequality) untouched because 7 is a positive number. Your final solution is x > 1. The answer says that any number larger than one can replace the x’s in the original inequality and make the inequality into a true statement.

The rules for solving linear equations (see the section “Linear Equations: Handling the First Degree”) also work with inequalities – somewhat. Everything goes smoothly until you try to multiply or divide each side of an inequality by a negative number.

When you multiply or divide each side of an inequality by a negative number, you have to reverse the sense (change < to >, or vice versa) to keep the inequality true.

The inequality 4(x – 3) – 2> 3(2x + 1) + 7, for example, has grouping symbols that you have to deal with. Distribute the 4 and 3 through their respective multipliers to make the inequality into 4x – 12 – 2> 6x + 3 + 7. Simplify the terms on each side to get 4x – 14 > 6x + 10. Now you put your inequality skills to work. Subtract 6x from each side and add 14 to each side; the inequality becomes –2x> 24. When you divide each side by –2, you have to reverse the sense; you get the answer x< – 12. Only numbers smaller than –12 or exactly equal to –12 work in the original inequality.

When solving the previous example, you have two choices when you get to the step 4x – 14> 6x + 10, based on the fact that the inequality a < b is equivalent to b > a. If you subtract 6x from both sides, you end up dividing by a negative number. If you move the variables to the right and the numbers to the left, you don’t have to divide by a negative number, but the answer looks a bit different. If you subtract 4x from each side and subtract 10 from each side, you get –24> 2x. When you divide each side by 2, you don’t change the sense, and you get –12> x. You read the answer as “–12 is greater than or equal to x.” This inequality has the same solutions as x< – 12, but stating the inequality with the number coming first is a bit more awkward.

Introducing interval notation

You can alleviate the awkwardness of writing answers with inequality notation by using another format called interval notation. You use interval notation extensively in calculus, where you’re constantly looking at different intervals involving the same function. Much of higher mathematics uses interval notation, although I really suspect that book publishers pushed its use because it’s quicker and neater than inequality notation. Interval notation uses parentheses, brackets, commas, and the infinity symbol to bring clarity to the murky inequality waters.

And, surprise surprise, the interval-notation system has some rules:

✔ You order any numbers used in the notation with the smaller number to the left of the larger number.

✔ You indicate “or equal to” by using a bracket.

✔ If the solution doesn’t include the end number, you use a parenthesis.

✔ When the interval doesn’t end (it goes up to positive infinity or down to negative infinity), use +∞ or –∞, whichever is appropriate, and a parenthesis.

Here are some examples of inequality notation and the corresponding interval notation:

Notice that the second example has a bracket by the –2, because the “greater than or equal to” indicates that you include the –2 also. The same is true of the 4 in the third example. The last example shows you why interval notation can be a problem at times. Taken out of context, how do you know if (–3, 7) represents the interval containing all the numbers between –3 and 7 or if it represents the point (–3, 7) on the coordinate plane? You can’t tell. You consider the context. A problem containing such notation has to give you some sort of hint as to what it’s trying to tell you.

Compounding inequality issues

A compound inequality is an inequality with more than one comparison or inequality symbol – for instance, –2 < x < 5. To solve compound inequalities for the value of the variables, you use the same inequality rules (see the intro to this section), and you expand the rules to apply to each section (intervals separated by inequality symbols).

To solve the inequality , for example, you add 5 to each of the three sections and then divide each section by 3:

Ancient symbols for timeless operations

Many ancient cultures used their own symbols for mathematical operations, and the cultures that followed altered or modernized the symbols for their own use. You can see one of the first symbols used for addition in the following figure, located on the far left – a version of the Italian capital P for the word piu, meaning plus. Tartaglia, a self-taught 16th century Italian mathematician, used this symbol for addition regularly. The modern plus symbol, +, is probably a shortened form of the Latin word et, meaning and.

The second figure from the left is what Greek mathematician Diophantes liked to use in ancient Greek times for subtraction. The modern subtraction symbol, –, may be a leftover from what the traders in medieval times used to indicate differences in product weights.

Leibniz, a child prodigy from the 17th century who taught himself Latin, preferred the third symbol from the left for multiplication. One modern multiplication symbol, × or , is based on St. Andrew’s Cross, but Leibniz used the open circle because he thought that the modern symbol looked too much like the unknown x.

The symbol on the far right is a somewhat backward D, used in the 18th century by French mathematician Gallimard for division. The modern division symbol,÷ , may come from a fraction line with dots added above and below.

You write the answer, , in interval notation as [–1, 5).

Here’s a more complicated example. You solve the problem by subtracting 5 from each section and then dividing each section by –2. Of course, dividing by a negative means that you turn the senses around:

You write the answer, , backward as far as the order of the numbers on the number line; the number –1 is smaller than 3. To flip the inequality in the opposite direction, you reverse the inequalities, too:. In interval notation, you write the answer as [–1, 3)