The solution set of inequality log∣x+1∣≥log3+log∣2x−1∣ is....
Explanation
The logarithm conditions are
∣x+1∣>0→x=−1
∣2x−1∣>0→x=21
Recall some basic concepts of absolute value inequality
logaf(x)≥logag(x)→f(x)≥g(x)
With condition a>1.
Let's solve the inequality
log∣x+1∣≥log3+log∣2x−1∣
log∣x+1∣≥log3∣2x−1∣
log∣x+1∣≥log∣6x−3∣
∣x+1∣≥∣6x−3∣
The solution
[(x+1)+(6x−3)][(x+1)−(6x−3)]≥0
(7x−2)(−5x+4)≥0
x=72∨x=54
The number line
Solution Number Line
Interval [72,54] indicates both endpoints are included in the solution set.
−
+
−
72
54
Therefore the solution is
{72≤x≤54,x=21}