Question

Error Analysis: Determine which step the first mistake was made in.
Solve for x:
x
+
1
x
−
2
−
3
x
+
4
=
0
Step 1:
Get a common denominator
(
x
+
4
)
(
x
+
1
)
(
x
+
4
)
(
x
−
2
)
−
3
(
x
−
2
)
x
+
4
(
x
−
2
)
=
0
Step 2:
Expand and simplify
x
2
+
5
x
+
4
−
3
x
−
6
(
x
+
4
)
(
x
−
2
)
=
0
Step 3:
Combine like terms
x
2
+
2
x
−
2
(
x
+
4
)
(
x
−
2
)
=
0
Step 4:
Set the numerator equal to zero and use the quadratic formula to solve.
x
2
+
2
x
−
2
=
0
x
=
−
2
±
√
4
−
8
2
Step 5:
Simplify
x
=
−
2
±
√
−
4
2
=
−
2
±
2
i
2
=
−
1
±
i
(10 points)
Step 1
Step 2
Step 3
Step 4
Step 5

Answer 1

Step 1.

Why: The LCD is (x−2)(x+4). After combining the fractions you should get

(x+1)(x+4) − 3(x−2) = 0,

or equivalently multiply both sides by (x−2)(x+4) to obtain that equation. The work shown in Step 1 has extra/misapplied factors and is not the correct numerator.

Correct solution sketch:
1. (x+1)/(x−2) − 3/(x+4) = 0
2. (x+1)(x+4) − 3(x−2) = 0
3. x^2+5x+4 −3x+6 = 0 → x^2+2x+10 = 0
4. x = [−2 ± √(4−40)]/2 = (−2 ± 6i)/2 = −1 ± 3i

(Exclude x = 2, −4 from the domain.)