Solve each of the following in equations and represent the solution set on the number line.
< 2, x ϵ R.
Given:
< 2, x ϵ R.
Intervals of |x - 3|
|x – 3| = -(x – 3) or (x – 3)
When |x - 3| = x – 3
x – 3 ≥ 0
Therefore, x ≥ 3
When |x - 3| = -(x – 3)
(x – 3) < 0
Therefore, x < 3
Intervals: x ≥ 3 or x < 3
Domain of < 2:
is not defined for x = 0
Therefore, x > 0 or x < 0
Now, combining intervals and domain:
x < 0 or 0 < x < 3 or x ≥ 3
For x = 0
→
Now, subtracting 2 from both the sides
Signs of 3 – 4x:
3 – 4x = 0 →
(Subtracting 3 from both the sides and then dividing both sides by -1)
3 – 4x > 0 →
(Subtracting 3 from both the sides and then multiplying both sides by -1)
3 – 4x < 0 →
(Subtracting 3 from both the sides and then multiplying both sides by -1)
Signs of x:
x = 0
x < 0
x > 0
Intervals satisfying the required condition: < 0
x < 0 or
Combining the intervals:
x < 0 or and x < 0
Merging the overlapping intervals:
x < 0
Similarly, for 0 < x < 3
x < 0 or and 0 < x < 3
Merging the overlapping intervals:
< x < 3
For, x ≥ 3
→
Now, subtracting 2 from both the sides
Signs of -3 – 2x:
-3 – 2x = 0 →
(Adding 3 to both the sides and then dividing both sides by -2)
-3 – 2x > 0 →
(Adding 3 to both the sides and then multiplying both sides by -1)
-3 – 2x < 0 →
(Adding 3 to both the sides and then multiplying both sides by -1)
Signs of x:
x = 0
x < 0
x > 0
Intervals satisfying the required condition: < 0
or x > 0
Combining the intervals:
or x > 0 and x ≥ 3
Merging the overlapping intervals:
x ≥ 3
Combining all the intervals:
x < 0 or or x ≥ 3
Merging overlapping intervals:
x < 0 and
Therefore,