Answer :
[tex] \frac{3x-5}{x-5} [/tex] > 0
First, note that x is undefined at 5. / x ≠ 5
Second, replace the inequality sign with an equal sign so that we can solve it like a normal equation. / Your problem should look like: [tex] \frac{3x-5}{x-5} [/tex] = 0
Third, multiply both sides by x - 5. / Your problem should look like: 3x - 5 = 0
Forth, add 5 to both sides. / Your problem should look like: 3x = 5
Fifth, divide both sides by 3. / Your problem should look like: x = [tex] \frac{5}{3} [/tex]
Sixth, from the values of x above, we have these 3 intervals to test:
x < [tex] \frac{5}{3} [/tex]
[tex] \frac{5}{3} [/tex] < x < 5
x > 5
Seventh, pick a test point for each interval.
1. For the interval x < [tex] \frac{5}{3} [/tex] :
Let's pick x - 0. Then, [tex] \frac{3x0-5}{0-5} [/tex] > 0
After simplifying, we get 1 > 0 which is true.
Keep this interval.
2. For the interval [tex] \frac{5}{3} [/tex] < x < 5:
Let's pick x = 2. Then, [tex] \frac{3x2-5}{2-5} [/tex] > 0
After simplifying, we get -0.3333 > 0, which is false.
Drop this interval.
3. For the interval x > 5:
Let's pick x = 6. Then, [tex] \frac{3x6-5}{6-5} [/tex] > 0
After simplifying, we get 13 > 0, which is ture.
Keep this interval.
Eighth, therefore, x < [tex] \frac{5}{3} [/tex] and x > 5
Answer: x < [tex] \frac{5}{3} [/tex] and x > 5
First, note that x is undefined at 5. / x ≠ 5
Second, replace the inequality sign with an equal sign so that we can solve it like a normal equation. / Your problem should look like: [tex] \frac{3x-5}{x-5} [/tex] = 0
Third, multiply both sides by x - 5. / Your problem should look like: 3x - 5 = 0
Forth, add 5 to both sides. / Your problem should look like: 3x = 5
Fifth, divide both sides by 3. / Your problem should look like: x = [tex] \frac{5}{3} [/tex]
Sixth, from the values of x above, we have these 3 intervals to test:
x < [tex] \frac{5}{3} [/tex]
[tex] \frac{5}{3} [/tex] < x < 5
x > 5
Seventh, pick a test point for each interval.
1. For the interval x < [tex] \frac{5}{3} [/tex] :
Let's pick x - 0. Then, [tex] \frac{3x0-5}{0-5} [/tex] > 0
After simplifying, we get 1 > 0 which is true.
Keep this interval.
2. For the interval [tex] \frac{5}{3} [/tex] < x < 5:
Let's pick x = 2. Then, [tex] \frac{3x2-5}{2-5} [/tex] > 0
After simplifying, we get -0.3333 > 0, which is false.
Drop this interval.
3. For the interval x > 5:
Let's pick x = 6. Then, [tex] \frac{3x6-5}{6-5} [/tex] > 0
After simplifying, we get 13 > 0, which is ture.
Keep this interval.
Eighth, therefore, x < [tex] \frac{5}{3} [/tex] and x > 5
Answer: x < [tex] \frac{5}{3} [/tex] and x > 5