Step-by-step explanation:
5x
−
3
4x−1
<1
First, find a common denominator for the fractions, which is 12:
5
⋅
3
4
⋅
3
−
4
−
1
⋅
4
3
⋅
4
<
1
4⋅3
5x⋅3
−
3⋅4
4x−1⋅4
<1
This simplifies to:
15
12
−
4
(
4
−
1
)
12
<
1
12
15x
−
12
4(4x−1)
<1
Next, simplify the fractions:
15
12
−
16
−
4
12
<
1
12
15x
−
12
16x−4
<1
Combine the fractions:
15
−
(
16
−
4
)
12
<
1
12
15x−(16x−4)
<1
15
−
16
+
4
12
<
1
12
15x−16x+4
<1
−
+
4
12
<
1
12
−x+4
<1
Multiply both sides by 12 to clear the fraction:
−
+
4
<
12
−x+4<12
Subtract 4 from both sides:
−
<
8
−x<8
Multiply both sides by -1 (remember to reverse the inequality):
>
−
8
x>−8
So, the solution to the inequality is
>
−
8
x>−8. In interval notation, this is
(
−
8
,
∞
)
(−8,∞).
The correct answer is:
(c)
(
8
,
∞
)
(8,∞)