Linear Inequalities and Absolute Value

Inequalities

Section P-9

Objectives

Use interval notation. Find intersections & unions of intervals. Solve linear inequalities. Recognize inequalities with no solution or all

numbers as solutions. Solve compound inequalities. Solve absolute value inequalities.

Interval Notation

Parentheses indicate endpoints that are not included in an interval.

Square brackets indicate endpoints that are included in an interval.

Interval Notation

Interval notation: used to represent solution sets open interval: use parenthesis ( ); use with <

or >; does not include endpoint closed interval: use brackets [ ]; use with < or

> ; includes endpoint Note: An interval can be half-open.

Examples

Write in interval notation and graph.

a) x < -1 b) x > -3 c) -4 < x < 2

Linear Inequalities

For equalities, you are finding specific values that will make your expression EQUAL something.

For inequalities, you are looking for values that will make your expression LESS THAN (or equal to), or MORE THAN (or equal to) something.

In general, your solution set will involve an interval of values that will make the expression true, not just specific points.

Properties of Inequalities…

are the same as with equations…

Use the GOLDEN RULEGOLDEN RULE.

Note: Always write an inequality with the variable on the left.

Especially before graphing.

EXAMPLES

See Example 4, page 114.

Solve and graph the solution set on a number line:

3x + 1 > 7x - 15

Compound Inequalities

We can write

-3 < 2x + 1 and 2x + 1 < 3

without the word “and” as a compound inequality. -3 < 2x + 1 < 3

To solve, perform the same operations on all three parts of the inequality and isolate the variable in the middle.

See Example 5, page 114-115.

Solve and graph the solution set on a number line:

1 < 2x + 3< 11

What if you have more than one inequality?

If two inequalities are joined by the word “AND”, you are looking for values that will make BOTH true at the same time. (the INTERSECTION of the 2 sets)

If two inequalities are joined by the word “OR”, you are looking for values that will make one inequality OR the other true (not necessarily both), therefore it is the UNION of the 2 sets.

EXAMPLE

Absolute Value Review

Absolute value means the distance of x from zero on a number line.

Symbol for absolute value is | |. Evaluate: |3| = |-5| =

- |17| -| -6 | =

What IS an absolute value inequality?

IF the absolute value expression is LESS THAN a value, you’re looking for values that are WITHIN that distance (intersection of the 2 inequalities).

IF the absolute value expression is MORE THAN a value, you’re looking for values that are getting further away in both directions (union of the 2 inequalities).

See Example 6, pg. 115. To solve |ax+b|<k, set up & solve

NIX: ax+b < k and ax+b > -k

BOOK: -k < ax+b < k

Solve |x - 2| < 5.

If, however, the absolute value was LESS than a number (think of this as a distance problem), you’re getting closer to your value and staying WITHIN a certain range. Therefore, this is an intersection problem (AND).

Example:

See Example 7, pg. 116.

To solve |ax+b|>k, set up & solve

ax+b < -k or ax+b > k.

Solve -3|5x – 2 | + 20 > -19.

If the absolute value is greater than a number, you’re considering getting further away in both directions, therefore an OR. (get further away left OR right)

Example:

EXAMPLES

Don’t leave common sense at the door!

Remember to use logic! Can an absolute value ever be less than or

equal to a negative value?? NO! (therefore if such an inequality were presented, the solution would be the empty set)

Can an absolute value ever be more than or equal to a negative value?? YES! ALWAYS! (therefore if such an inequality were given, the solution would be all real numbers)