Chapter 1: Section 5 Notes
By the end of the section, you should be able to:
- Solve inequalities and compound inequalities.
- Visually represent (graph) inequalities and compound inequalities on a number line.
- Create inequalities and compound inequalities to model situations.
Definitions:
- Inequality: a relationship between two expressions when one expression is greater than (>), greater than or equal to (≥), less than (<), or less than or equal to (≤) the other expression.
- Compound Inequality: two inequalities that are joined with the word and or the word or.
Properties of Inequalities
Solving Inequalities
Things/Steps to Remember:
- Greater than (>) and Less than (<) inequalities are graphed with open circles (○).
- Greater than or equal to (≥) and Less than or equal to (≤) inequalities are graphed with closed circles (●).
- The arrow is extended to the right for graphs of greater than and greater than or equal to inequalities.
- The arrow is extended to the left for graphs of less than and less than or equal to inequalities.
Always/Never Inequalities
Things/Steps to Remember:
- If the inequality simplifies to a statement that is never true such as: , the equation has no solution.:
- If the inequality simplifies to an a statement that is always true such as: , the equation has infinite solutions.
- If the inequality simplifies to a statement that is sometimes true such as: , then the only solution is the value that makes it true (which is true only when x has the value less than or equal to 3).
Solving Compound Inequalities
Things/Steps to Remember:
- An and compound inequality means that a solution makes BOTH inequalities true.
- Ex. For the inequalities x < 3 and x > 1, only numbers that are less than 3 and greater than 1 are solutions. Possible solutions are x =2, x=2.75, x = 1.5, etc.
- Ex. For the inequalities x > 3 and x < 1, only numbers that are greater than 3 and less than 1 are solutions. There are no possible solutions that would make both inequalities true. This compound inequality would have no solution.
- An or compound inequality means that a solution makes EITHER inequalities true.
- Ex. For the inequalities x < 3 or x > 1, only numbers that are less than 3 or greater than 1 are solutions. Any number for x would satisfy one or both of the inequalities. This compound inequality would have infinite solutions.
- Ex. For the inequalities x > 3 or x < 1, only numbers that are greater than 3 or less than 1 are solutions. The solutions to this compound inequality would be all numbers EXCEPT numbers between 1 and 3 because they would not make either inequality true.
BONUS video: Why do we flip the inequality when multiplying/dividing by a negative number?
Section 1.5 Worksheet |
Section 1.5 WS Answers |
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