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Warm Up
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Algebra 1 Book Pg 352 # 1, 4, 7, 10, 12
Warm Up Answers
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Algebra 1 Book Pg 352 # 1, 4, 7, 10, 12
1. x < 5
4. x < -10
7. x < 2
10. x > -9 or x < -12
12. x < -1 or x > 29/5
Solving Linear Inequalities
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Algebra 1 Textbook section 6.4
Algebra 2 Textbook section 1.7
The student will be able to:
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Solve and graph absolute value equations
Solve and graph absolute value inequalities
Solving Absolute Value Equations
• Absolute value is denoted by bars on
each side of a number or expression |3|.
• Absolute value represents the distance a
number is from 0. Thus, it is always
positive.
• |8| = 8 and |-8| = 8
Solving absolute value equations
• First, isolate the absolute value expression.
• Set up two equations to solve.
• Same and Opposite
• For the first equation, write it the Same way you see it
without the absolute value bars and solve.
• For the second equation, write it the Opposite way without
the absolute value bars and solve
• Only write the opposite of the RIGHT SIDE of the equal
sign.
• Always check the solutions.
6|5x + 2| = 312
• Isolate the absolute value expression by dividing by 6.
6|5x + 2| = 312
|5x + 2| = 52
Same
5x + 2 = 52
5x = 50
x = 10
Opposite
or
5x + 2 = -52
5x = -54
x = -10.8
•Check: 6|5x + 2| = 312
6|5(10)+2| = 312
6|52| = 312
312 = 312
6|5x + 2| = 312
6|5(-10.8)+ 2| = 312
6|-52| = 312
312 = 312
3|x + 2| -7 = 14
• Isolate the absolute value expression by adding 7 and dividing
by 3.
3|x + 2| -7 = 14
3|x + 2| = 21
|x + 2| = 7
Same
x+2=7
x=5
Opposite
or
•Check: 3|x + 2| - 7 = 14
3|5 + 2| - 7 = 14
3|7| - 7 = 14
21 - 7 = 14
14 = 14
x + 2 = -7
x = -9
3|x + 2| -7 = 14
3|-9+ 2| -7 = 14
3|-7| -7 = 14
21 - 7 = 14
14 = 14
Review of the Steps to Solve a
Compound Inequality:
● Example: 2 x  3  2 and 5 x  10
● You must solve each part of the inequality.
● The graph of the solution of the “and”
statement is the intersection of the two
inequalities. Both conditions of the inequalities
must be met.
●In other words, the solution is wherever
the two inequalities overlap.
●If the solution does not overlap, there is
no solution.
Review of the Steps to Solve a
Compound Inequality:
● Example: 3x  15 or -2x +1  0
● You must solve each part of the inequality.
● The graph of the solution of the “or” statement is
the union of the two inequalities. Only one
condition of the inequality must be met.
● In other words, the solution will include each
of the graphed lines. The graphs can go in
opposite directions or towards each other, thus
overlapping.
● If the inequalities do overlap, the solution is all
real numbers.
Solving an Absolute Value Inequality
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Step 1: Rewrite the inequality as an AND or an OR
statement :
● If you have a  or  you are working with an
‘and’ statement.
Remember: “Less thand”
● If you have a  or  you are working with an
‘or’ statement.
Remember: “Greator”
Step 2: In the second equation you must negate the
right hand side and reverse the direction of the
inequality sign. Set up a SAME inequality and an
OPPOSITE inequality
Solve as a compound inequality.
Example 1:
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This is an ‘or’ statement.
(Greator). Rewrite.
|2x + 1| > 7
2x + 1 > 7 or 2x + 1 >7
2x + 1 >7 or 2x + 1 <-7
x > 3 or
In the 2nd inequality, reverse the
inequality sign and negate the
right side value.
Solve each inequality.
x < -4
Graph the solution.
-4
3
Example 2:
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|x -5|< 3
This is an ‘and’ statement.
(Less thand).
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x -5< 3 and x -5< 3
x -5< 3 and x -5> -3
Rewrite.
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In the 2nd inequality, reverse the
inequality sign and negate the
right side value.
x < 8 and x > 2
2<x<8
Solve each inequality.
Graph the solution.
2
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Charts to Help
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Copy the chart from the following page
in your notes!
Algebra 1 Pg 354
Algebra 2 Pg 53