Linear Inequalities in One Variable Objective: To solve linear inequalities.
SECTION 1-4: Solving Inequalities We solve inequalities the same way we solve equations with the...
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Transcript of SECTION 1-4: Solving Inequalities We solve inequalities the same way we solve equations with the...
SECTION 1-4: Solving Inequalities
We solve inequalities the same way we solve equations with the following
exception:
**GOLDEN RULE for Inequalities**
When you multiply or divide by a __________ number, you MUST _______ the direction of the inequality symbol.
The Inequality Symbols
Key words that describe each symbol:1. < - less than,
2. > - greater than,
3. ≤ - less than or equal to,
4. ≥ - greater than or equal to,
Solving Inequalities - EXAMPLES
EX.A) -3x > 6 B) 2 - x ≤ -79
1
Graphing Solutions of Inequalities Rules for graphing inequalities:
< or > - use an ________ dot≤ or ≥ - use a _________ dot
< or ≤ - shade to the ________> or ≥ - shade to the ________
** The variable must be on the _______ after you solve to use these rules!! (Ex. x < 3)
Graph the solution: EXAMPLES
EX.A) 3x – 12 < 3
Graph: ------------------------------------
Is ___ part of the solution?
Check your answer
How can we check our answer to EX.A if 5 is not part of the solution??
EXAMPLES – Graphing the Solution
EX.B) 9 – 2x > 5 EX.C) 3x – 7 ≤ 5
---------------------- ----------------------
ALL REAL NUMBERS & NO SOLUTION
When our result has no variable left in it, our answer is either all real numbers or no solution.
If the result is _______ (Ex. 3 < 7), our answer is ________________________________.
If the result is _______ (Ex. 3 > 7), our answer is ________________________________.
EXAMPLES
EX.1) 2x – 3 > 2(x – 5)
Our result is ______. Therefore, our answer is ___________________________.
Graph: ----------------------------------
EXAMPLES
EX.2) 7x + 6 < 7(x – 4)
Our result is ______, therefore our answer is _______________________.
Graph: --------------------------------
EXAMPLES – Try These:
1) 2x < 2(x + 1) + 3
2) 4(x – 3) + 7 ≥ 4x + 1
3) 4x + 8 > -4(x – 8)
INEQUALITY WORD PROBLEMS - write an inequality for the situation
EX. A band agrees to play for $200 plus 25% of the ticket sales. Find the ticket sales needed for the band to receive at least $500.
Define variables: Let x = __________________
In words, $200 + 25% ticket sales _______ $500
Write an inequality:
Inequality word problems…
Solve the inequality:
Write a sentence for your answer: ______________________________________________________________________________________________________________________________
Inequality word problems…Example 2
A salesperson earns a salary of $700 per month plus 2% of the sales. What must the sales be if the salesperson is to have monthly income of at least $1800.
Let x = _____________________________
Write an equation:
Example 2, continued…
Solve the inequality:
Write a sentence for your answer: _________
_____________________________________________________________________________________________________________________
Example 3
The lengths of the sides of a triangle are 3:4:5. What is the length of the longest side if the perimeter is not more than 84 cm?
Use x to represent the ratio. s1 =
s2 =
s3 =
Example 3, continued…
Write an inequality from the given information:
What is the length of the
longest side??
COMPOUND INEQUALITIES
Compound inequalities are ________ of inequalities joined by _______ or ________.
If ‘and’ and ‘or’ are not written, use the following rule:
Less thAN (<, ≤) use ANd
GreatOR (>, ≥) use OR
‘AND’ Graphs
AND represents the overlap, also called the ___________ of the two inequalities.
We need to transfer everything with 2 lines above onto our final graph.
EX. -----------------------------------
EX. -----------------------------------
‘AND’ Examples
3x – 1 > -28 AND 2x + 7 < 19STEP 1: Solve each inequality separately
Step 2: Graph each above the final number line
Step 3: ----------------------------------
‘AND’ Examples
2x < x + 6 < 4x – 18 (less thAN use AND)
STEP 1: Solve each inequality separately
Step 2: Graph each above the final number line
Step 3: ----------------------------------
‘OR’ Graphs
OR represents the ________ of the two inequalities.
We need to transfer everything with 1 or more lines above onto our final graph.
EX. -----------------------------------
EX. -----------------------------------
‘OR’ Examples
4y – 2 ≥ 14 OR 3y – 4 ≤ -13STEP 1: Solve each inequality separately
Step 2: Graph each above the final number line
Step 3: ----------------------------------
‘OR’ Examples
x - 12 ≥ -5x ≥ -2x – 9 (greatOR use OR)STEP 1: Solve each inequality separately
Step 2: Graph each above the final number line
Step 3: ----------------------------------