Solve for x:
A|x + E| + B =
C|x + E| + D
x =
or
x =
Subtract
\red{abs(C)|x + E|}
from both sides:
Add
\red{abs(C)|x + E|}
to both sides:
\qquad\begin{eqnarray}
A|x + E| + B
&=&
C|x + E| + D
\\ \\
\red{ - C|x + E|}
&&
\red{ - C|x + E|} \\ \\
A - C|x + E| +
B
&=& D
\end{eqnarray}
Subtract
\red{abs(B)}
from both sides:
Add
\red{abs(B)}
to both sides:
\qquad\begin{eqnarray}
A - C|x + E| +
B &=& D \\ \\
\red{ - B} &&
\red{ - B} \\ \\
A - C|x + E| &=&
D - B
\end{eqnarray}
Divide both sides by
\red{A - C}:
\qquad
\dfrac{A - C|x + E|}
{\red{A - C}} =
\dfrac{D - B}
{\red{A - C}}
Simplify:
\qquad |x + E| =
SIMPLIFIED
Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive:
\qquad
x + E = -SIMPLIFIED
or
\qquad
x + E = SIMPLIFIED
Solve for the solution where
x + E is negative:
\qquad
x + E = -SIMPLIFIED
Subtract
\red{abs(E)}
from both
sides:
Add
\red{abs(E)}
to both
sides:
\qquad\begin{eqnarray}
x + E &=&
-SIMPLIFIED \\ \\
\red{- E} &&
\red{- E} \\ \\
x &=& -SIMPLIFIED -
E
\end{eqnarray}
Change the
\red{{} - E}
to an equivalent fraction with a
denominator of
SIMPLIFIED_DENOM:
\qquad
x = - SIMPLIFIED
\red{E > 0 ? "-" : "+"
fraction(abs(E) * SIMPLIFIED_DENOM,
SIMPLIFIED_DENOM)}
\qquad
x = fractionReduce.apply(null,
NEG_SOLUTION)
Then calculate the solution where
x + E is positive:
\qquad
x + E = SIMPLIFIED
Subtract
\red{abs(E)}
from both
sides:
Add
\red{abs(E)}
to both
sides:
\qquad\begin{eqnarray}
x + E &=&
SIMPLIFIED \\ \\
\red{- E} &&
\red{- E} \\ \\
x &=& SIMPLIFIED -
E
\end{eqnarray}
Change the
\red{{} - E}
to an equivalent fraction with a
denominator of
SIMPLIFIED_DENOM:
\qquad
x = SIMPLIFIED
\red{E > 0 ? "-" : "+"
fraction(abs(E) * SIMPLIFIED_DENOM,
SIMPLIFIED_DENOM)}
\qquad
x = fractionReduce.apply(null,
POS_SOLUTION)
Subtract
\red{A|x + E|}
from both sides:
Add
\red{A|x + E|}
to both sides:
\qquad\begin{eqnarray}
A|x + E| + B
&=&
C|x + E| + D
\\ \\ \red{- A|x + E|}
&&
\red{- A|x + E|} \\ \\
B &=&
C - A|x + E| +
D
\end{eqnarray}
Subtract
abs(D)
from both sides:
Add
abs(D)
to both sides:
\qquad\begin{eqnarray}
B &=&
C - A|x + E| +
D \\ \\
\red{- D} &&
\red{- D} \\ \\
B - D &=&
C - A|x + E|
\end{eqnarray}
Divide both sides by
\red{C - A}.
\qquad
\dfrac{B - D}
{\red{C - A}} =
\dfrac{C - A|x + E|}
{\red{C - A}}
Simplify:
\qquad
SIMPLIFIED = |x + E|
Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive:
\qquad
-SIMPLIFIED = x + E
or
\qquad
SIMPLIFIED = x + E
Solve for the solution where
x + E is negative:
\qquad - SIMPLIFIED = x +
E
Subtract
\red{abs(E)}
from both
sides:
Add
\red{abs(E)}
to both
sides:
\qquad\begin{eqnarray}
- SIMPLIFIED &=&
x + E \\ \\
\red{- E} &&
\red{- E} \\ \\
-SIMPLIFIED - E
&=& x
\end{eqnarray}
Change the
\red{{} - E}
to an equivalent fraction with a
denominator of
SIMPLIFIED_DENOM.
\qquad
- SIMPLIFIED
\red{E > 0 ? "-" : "+"
fraction(abs(E) * SIMPLIFIED_DENOM,
SIMPLIFIED_DENOM)} = x
\qquad
fractionReduce.apply(null,
NEG_SOLUTION) = x
Then calculate the solution where
x + E is positive:
\qquad
SIMPLIFIED = x + E
Subtract
\red{abs(E)}
from both
sides:
Add
\red{abs(E)}
to both
sides:
\qquad\begin{eqnarray}
SIMPLIFIED &=&
x + E \\ \\
\red{- E} &&
\red{- E} \\ \\
SIMPLIFIED - E
&=& x
\end{eqnarray}
Change the
\red{{} - E}
to an equivalent fraction with a
denominator of
SIMPLIFIED_DENOM.
\qquad
SIMPLIFIED
\red{E > 0 ? "-" : "+"
fraction(abs(E) * SIMPLIFIED_DENOM,
SIMPLIFIED_DENOM)} = x
\qquad
fractionReduce.apply(null,
POS_SOLUTION) = x
Thus, the correct answer is
x =
fractionReduce.apply(null, NEG_SOLUTION)
or
x =
fractionReduce.apply(null, POS_SOLUTION)
.
The absolute value cannot be negative. Therefore, there is no solution.