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Title: Absolute value equations and inequalities (Algebra)
Description: Absolute value is a mathematical function that returns the distance of a number from zero on a number line. In algebra, we use absolute value equations and inequalities to solve problems where we need to find the value(s) of a variable that satisfy a given condition involving absolute value. An absolute value equation involves an absolute value expression set equal to a constant or another expression. To solve an absolute value equation, we need to consider two cases: one where the expression inside the absolute value is positive, and one where it's negative. We solve each case separately and then combine the solutions.
Description: Absolute value is a mathematical function that returns the distance of a number from zero on a number line. In algebra, we use absolute value equations and inequalities to solve problems where we need to find the value(s) of a variable that satisfy a given condition involving absolute value. An absolute value equation involves an absolute value expression set equal to a constant or another expression. To solve an absolute value equation, we need to consider two cases: one where the expression inside the absolute value is positive, and one where it's negative. We solve each case separately and then combine the solutions.
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1
...
8 - 1
Distance
is 3
...
Distance
is 3
...
than 3
...
–3
0
3
By definition, the equation |x| = 3 can be
solved by finding real numbers at a distance
of three units from 0
...
So the solution set is {−3,3}
...
8 - 2
Properties of Absolute Value
1
...
2
...
For any positive number b:
3
...
4
...
1
...
5 − 3 x =
12
Solution
For the given expression 5 – 3x to have
absolute value 12, it must represent either 12
or –12
...
1
...
5 − 3 x =
12
Solution
5 − 3x =
12
5 − 3x =
12 or
−3 x =
7
7
x= −
3
or
or
5 − 3x =
−12
Property 1
−3 x =
−17
Subtract 5
...
1
...
5 − 3 x =
12
Solution
17
7
or
x=
x= −
Divide by –3
...
The
solution set is
7 17
...
8 - 6
Example 1
SOLVING ABSOLUTE VALUE
EQUATIONS
Solve
b
...
x = −
5
5
x =3
{ }
1
...
2 x + 1 < 7
Solution
Use Property 3, replacing a with 2x + 1 and b
with 7
...
Divide each part by 2
...
8 - 8
Example 2
SOLVING ABSOLUTE VALUE
INEQUALITIES
Solve
a
...
The final inequality gives the solution set (–4, 3)
...
8 - 9
Example 2
SOLVING ABSOLUTE VALUE
INEQUALITIES
Solve
b
...
Divide each
part by 2
...
8 - 10
Example 2
SOLVING ABSOLUTE VALUE
INEQUALITIES
Solve
b
...
The solution set is ( − ∞, −4) ∪ (3, ∞ )
...
8 - 11
Example 3
SOLVING AN ABSOLUTE VALUE INEQUALITY
REQUIRING A TRANSFORMATION
Solve 2 − 7 x − 1 > 4
...
2 − 7x > 5
2 − 7 x < −5 or
−7 x < −7
x >1
or
or
2 − 7x > 5
Property 4
−7 x > 3
3
x<−
7
Subtract 2
...
1
...
Solution
x >1
or
3
x<−
7
Divide by –7;
reverse the
direction of each
inequality
...
7
1
...
2 − 5 x ≥ −4
Solution Since the absolute value of a
number is always nonnegative, the inequality
is always true
...
1
...
4 x − 7 < −3
Solution There is no number whose
absolute value is less than –3 (or less than
any negative number)
...
1
...
5 x + 15 =
0
Solution The absolute value of a number
will be 0 only if that number is 0
...
Check by
substituting into the original equation
...
8 - 16
Title: Absolute value equations and inequalities (Algebra)
Description: Absolute value is a mathematical function that returns the distance of a number from zero on a number line. In algebra, we use absolute value equations and inequalities to solve problems where we need to find the value(s) of a variable that satisfy a given condition involving absolute value. An absolute value equation involves an absolute value expression set equal to a constant or another expression. To solve an absolute value equation, we need to consider two cases: one where the expression inside the absolute value is positive, and one where it's negative. We solve each case separately and then combine the solutions.
Description: Absolute value is a mathematical function that returns the distance of a number from zero on a number line. In algebra, we use absolute value equations and inequalities to solve problems where we need to find the value(s) of a variable that satisfy a given condition involving absolute value. An absolute value equation involves an absolute value expression set equal to a constant or another expression. To solve an absolute value equation, we need to consider two cases: one where the expression inside the absolute value is positive, and one where it's negative. We solve each case separately and then combine the solutions.