Answer:
he first thing to do here is isolate the modulus on onse side of the equation by adding
4
a
to both sides
|
4
a
+
6
|
−
4
a
+
4
a
=
10
+
4
a
|
4
a
+
6
|
=
10
+
4
a
Now, by definition, the absolute value of a real number will only return positive values, regardless of the sign of said number.
This means that the first condition that any value of
a
must satisfy in order to be a valid solution will be
10
+
4
a
≥
0
4
a
≥
−
10
⇒
a
≥
−
5
2
Keep this in mind. Now, since the absolute value of a number returns a positive value, you can have two possibilities
4
a
+
6
<
0
⇒
|
4
a
+
6
|
=
−
(
4
a
+
6
)
In this case, the equation becomes
−
(
4
a
+
6
)
=
10
+
4
a
−
4
a
−
6
=
10
+
4
a
8
a
=
−
16
⇒
a
=
(
−
16
)
8
=
−
2
(
4
a
+
6
)
≥
0
⇒
|
4
a
+
6
|
=
4
a
+
6
This time, the equation becomes
4
a
+
6
=
10
+
4
a
6
≠
10
⇒
a
∈
∅
Therefore, the only valid solution will be
a
=
−
2
. Notice that it satisfies the initial condition
a
≥
−
5
2
.
Do a quick check to make sure that the calculations are correct
|
4
⋅
(
−
2
)
+
6
|
−
4
⋅
(
−
2
)
=
10
|
−
2
|
+
8
=
10
2
+
8
=
10
x
√
