College Algebra: Section 1.6 (Packet #1)
By the end of the section, you should be able to:
- Solve linear inequalities
- Solve compound linear inequalities
- Solve absolute value inequalities
- Translate inequality phrases
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
**For a more extensive review of Inequalities, please see Algebra 2 Section 1.5
**For a more extensive review of Inequalities, please see Algebra 2 Section 1.5
**The two symbols < and > are called strict inequality signs, while the symbols ≤ and ≥ are nonstrict inequality signs.
Example: a < b can be read as: “a is strictly less than b” meaning that a will never be equal/the same as b, it will always be less than b.
Example: a < b can be read as: “a is strictly less than b” meaning that a will never be equal/the same as b, it will always be less than b.
Solving Linear Inequalities
🎥Examples 1&2
🎥Examples 1&2
Things to remember about solving equations for different variables:
- Greater than(> and Less than < inequalities are graphed with open circles ○ or the curved brackets ) (
- Greater than or equal to ≥ and Less than or equal to ≤ inequalities are graphed with closed circles ● or square brackets ] ;
- 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.
Things to remember:
- An and compound inequality means that a solution makes BOTH inequalities true.
- In set notation an and would be the same operation as an intersection “∩“ of two intervals.
- 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. In interval notation (-∞ , 3) ∩ (1 , ∞) ⇒ (1 , 3).
- 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.
- In set notation an and would be the same operation as a union “∪“ of two intervals.
- 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. In interval notation (-∞ , 3) ∪ (1 , ∞) ⇒ (-∞ , ∞).
- 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.
Solving Absolute Value Inequalities
*These are videos from Algebra 2 Section 1.6 they should be labeled here as Examples 5&6
🎥Examples 3&4
*These are videos from Algebra 2 Section 1.6 they should be labeled here as Examples 5&6
🎥Examples 3&4
Things to remember about solving equations for different variables:
- ALWAYS get the Absolute Value by itself first, THEN break into two inequalities.
- If an absolute value is less than a number, then an AND compound inequality is formed.
- If an absolute value is greater than a number, then an OR compound inequality is formed.
Examples of inequality phrases:
- “x is no greater than y”
- This means that x is not greater than y, which is the same as saying x is less than or equal to y, so this translates to x ≤ y .
- This means that x is not greater than y, which is the same as saying x is less than or equal to y, so this translates to x ≤ y .
- “x is at least as large as y”
- If x is at least as large as y, then it can either be as large as (equal to) y or larger (greater) than y, so this phrase translates to x ≥ y.
- If x is at least as large as y, then it can either be as large as (equal to) y or larger (greater) than y, so this phrase translates to x ≥ y.
- “x does not exceed y”
- Compare this to the first phrase; the words “is no” carry the same meaning as “does not”, and “greater than” is a synonym for “exceed”. The two phrases have the same meaning, and so “x does not exceed y” also translates to x ≤ y.
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