MATH HELP: SOLVING INEQUALITIES?
Please help me with this problem:
I have tried so many ways but am still unable to solve it.
- llafferLv 71 month ago
This is similar to your other one.
(2x - 1)(x - 3)² / (x - 7) < 0
x can't be 7. if x > 7 the denominator is positive. if x < 7 the denominator is negative and will require us to flip the sign. Multiplying both sides by (x - 7) and splitting up the ranges of each inequlity into two gives us:
(2x - 1)(x - 3)² < 0 when x > 7 and (2x - 1)(x - 3)² > 0 when x < 7
in both cases (x - 3)² will always be positive so we only need to look at the other factor.
if x > 1/2 the product is positive. This is in the solution set of the second inequality
if x < 1/2 the product is negative. This is in the solution set of the first inequality
Since the inequalities are less than or greater than 0, then neither 1/2 nor 3 can be valid values for x. We'll keep this in mind for later.
Making this substitution into each we get:
x < 1/2 when x > 7 and x > 1/2 when x < 7
No values of x will be both greater than 7 and less than 1/2. So we can throw that out:
x > 1/2 when x < 7
There are values between 1/2 and 7 that will satisfy both of these so the answer is:
1/2 < x < 7
Remember when I said that x can't be 3? We have to take 3 out of it now to end up with this:
1/2 < x < 3 and 3 < x < 7
In interval notation: (1/2, 3) U (3, 7)
And in the graph, you would have open circles in three locations: 1/2, 3, and 7 and a lines connecting them between the first two and the last two.