Transcript Algebra II TE - Chapter 1 Section 4

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
1.
Key words that describe each symbol:
< - less than,
2.
> - greater than,
3.
≤ - less than or equal to,
4.
≥ - greater than or equal to,
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Solving Inequalities - EXAMPLES
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EX.A) -3x > 6
1
B) 2 - x ≤ -7
9
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
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EX.A) 3x – 12 < 3
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Graph: ------------------------------------
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Is ___ part of the solution?
Check your answer
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How can we check our answer to EX.A if 5 is
not part of the solution??
EXAMPLES – Graphing the Solution
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EX.B) 9 – 2x > 5
----------------------
EX.C) 3x – 7 ≤ 5
----------------------
ALL REAL NUMBERS & NO SOLUTION
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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
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EX.1) 2x – 3 > 2(x – 5)
Our result is ______. Therefore, our answer is
___________________________.
Graph:
----------------------------------
EXAMPLES
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EX.2) 7x + 6 < 7(x – 4)
Our result is ______, therefore our answer is
_______________________.
Graph:
--------------------------------
EXAMPLES – Try These:
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1) 2x < 2(x + 1) + 3
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2) 4(x – 3) + 7 ≥ 4x + 1
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3) 4x + 8 > -4(x – 8)
INEQUALITY WORD PROBLEMS
- write an inequality for the situation
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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…
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Solve the inequality:
Write a sentence for your answer: _________
_______________________________________
_______________________________________
_______________________________________
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Inequality word
problems…Example 2
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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.
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Let x = _____________________________
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Write an equation:
Example 2, continued…
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Solve the inequality:
Write a sentence for your answer: _________
_______________________________________
_______________________________________
_______________________________________
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Example 3
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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
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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
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AND represents the overlap, also called the
___________ of the two inequalities.
We need to transfer everything with 2 lines
above onto our final graph.
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EX.
-----------------------------------
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EX.
-----------------------------------
‘AND’ Examples
3x – 1 > -28 AND 2x + 7 < 19
STEP 1: Solve each inequality separately
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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
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Step 2: Graph each above the final number line
Step 3:
----------------------------------
‘OR’ Graphs
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OR represents the ________ of the two
inequalities.
We need to transfer everything with 1 or more
lines above onto our final graph.
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EX.
-----------------------------------
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EX.
-----------------------------------
‘OR’ Examples
4y – 2 ≥ 14 OR 3y – 4 ≤ -13
STEP 1: Solve each inequality separately
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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
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Step 2: Graph each above the final number line
Step 3:
----------------------------------