Chapter Four — Linear Equations in One Variable
The Humongous Book of Algebra Problems
70
Absolute Value Equations
Most of them have two solutions
4.29 What values of x satisfy the equation ?
This absolute value equation has two valid solutions, the value on the left side of
the equation and its opposite: x = 6 or x = –6. Substituting either for x results in
a true statement.
The solutions to the equation (where r is a positive real number) are
r and –r.
4.30 Solve the equation for x and verify the solutions.
Isolate the absolute value expression left of the equal sign by adding 3 to both
sides of the equation.
According to Problem 4.29, x = –4 or x = 4.
To verify the solutions, substitute each into the original equation and verify that
the results are true statements.
4.31 Explain why the equation has no real solutions.
The expression left of the equal sign, , is entirely enclosed by absolute
value bars. Therefore, the left side of the equation must be positive no matter
what value of x is substituted into it. The right side of the equation contains a
negative constant, –8. If the left side of the equation must be positive for all
real x and the right side of the equation is always negative, no x–value can be
substituted into the equation that results in a true statement.
Use the
word “or”
to separate
possible solutions,
because you get a
true statement by
plugging either x = 6
OR x = –6 into the
equation.
So whatever’s
inside the
absolute value
bars either equals
the left side of
the equation or the
opposite of the
left side.
After you
isolate the
absolute value
on the left side of
the equation, set
whatever’s inside the
bars (in this case x)
equal to the right side
of the equation (4)
and the opposite of
that number (–4).
Those are the
solutions.
Chapter Four — Linear Equations in One Variable
The Humongous Book of Algebra Problems
71
4.32 Solve the equation for x.
The absolute value expression is isolated on one side of the equation, so create
two new equations—one that sets the contents of the absolute value expression
equal to 12 and one that sets the same expression equal to –12.
The solution is x = –16 or x = 8.
4.33 Solve the equation for x.
The absolute value expression is isolated on one side of the equation, so create
two new equations—one that sets the contents of the absolute value expression
equal to 15 and one that sets the same expression equal to –15.
The solution is x = –6 or x = 9.
4.34 Solve the equation for x.
Isolate the absolute value expression left of the equal sign by subtracting 1 from
both sides of the equation.
Create two equations by setting the expression within the absolute value
symbols equal to 8 and its opposite, –8.
The solution is x = –3 or x = 13.
The rst
step is to get the
absolute values on one
side of the equal sign
and everything else on
the other side. In this
problem, the absolute
value bars are
already all alone on
the left side.
Chapter Four — Linear Equations in One Variable
The Humongous Book of Algebra Problems
72
4.35 Solve the equation for x.
To isolate the absolute value expression left of the equal sign, add 9 to both
sides of the equation.
The absolute value expression is not yet isolated because of its coefficient.
Divide both sides of the equation by 3.
Rewrite the statement as two equations that do not contain absolute values and
solve them.
The solution is or .
4.36 Solve the equation for x.
Isolate the absolute value expression on the left side of the equation.
Create two new equations using the technique described in Problem 4.32.
The solution is or .
This works
just like a
regular, nonabsolute
value, equation. To
solve 3x = 27 for x,
you’d divide both sides
by 3. If you’re isolating
something (either a
variable or an absolute
value expression), its
coefcient’s got
to go.
It’s the same
thing you’ve been
doing since Problem
4.30. Set whatever’s
inside the absolute
value bars equal to
the number on the
right side of the
equation. Then set
it equal to the
opposite of the
number on
the right.
Move the
constant to
the left side (by
adding 12) before
you eliminate the
absolute value
coefcient
.
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