linear inequalities

Linear Inequalities

To write that one number is greater than or less than another number, we use the following symbols.

INEQUALITY SYMBOLS

< means is less than

means is less than or equal to

> means is greater than

means is greater than or equal to

An equation states that two expressions are equal; an inequality states that they are unequal. A linear inequality is an inequality that can be simplified to the form ax < b (Properties introduced in this section are given only for <, but they are equally valid for >, or .) Linear inequalities are solved with the following properties.

PROPERTIES OF INEQUALITY

For all real numbers a, b, and c:

If a < b then a + c < b + c
If a < b and if c > 0 then ac < bc
If a < b and if c < 0 then ac > bc

Pay careful attention to property 3; it says that if both sides of an inequality are multiplied by a negative number, the direction of the inequality symbol must be reversed.

Solving Linear Inequalities

EXAMPLE

Solve 4 -3y 7 + 2y.

Solution

Use the properties of inequality.

4 -3y + (-4) 7 + 2y + (-4) Add -4 to both sides.

-3y 3 + 2y

Remember that adding the same number to both sides never changes the direction of the inequality symbol.

-3y + (-2y) 3 + 2y + (-2y) Add -2y to both sides.

-5y 3

Multiply both sides by -1/5. Since -1/5 is negative, change the direction of the inequality symbol.

CAUTION

It is a common error to forget to reverse the direction of the inequality sign when multiplying or dividing by a negative number. For example, to solve -4x 12 we must multiply by -1/4 on both sides and reverse the inequality symbol to get x -3.

The solution y -3/5 in the previous example represents an interval on the number line. Interval notation often is used for writing intervals. With interval notation, y -3/5 is written as . This is an example of a half-open interval, since one endpoint, -3/5, is included. The open interval (2, 5) corresponds to 2 < x < 5, with neither endpoint included. The closed interval [2, 5] includes both endpoints and corresponds to 2 x 5.

The graph of an interval shows all points on a number line that correspond to the numbers in the interval. To graph the interval , for example, use a solid circle at -3/5 since -3/5 is part of the solution. To show that the solution includes all real numbers greater than or equal to -3/5 draw a heavy arrowpointing to the right (the positive direction). See Figure 1.

Graphing Linear Inequalities

EXAMPLE

Solve -2 < 5 + 3m < 20. Graph the solution.

Solution

The inequality -2 < 5 + 3m < 20 says that 5 + 3m is between -2 and 20. Solve this inequality with an extension of the properties given above.Work as follows, first adding -5 to each part.

-2 + (-5) < 5 + 3m + (-5) < 20+ (-5)

-7 < 3m < 15

Now multiply each part by 1/3.

A graph of the solution is given in Figure 2; here open circles are used to show that -7/3 and 5 are not part of the graph.

(Some textbooks use brackets in place of solid circles for the graph of a closed interval, and parentheses in place of open circles for the graph of an open interval.)