solve the following inequality
15x-16 less than or equal to 15/x
is the point x=0 included in the solution set of the inquality?
are the other finite pointsa of the interval included in the solution set?
what is the solution set?
please show work
Look at how I did the one for Krystal right above this question.
To solve the inequality 15x - 16 ≤ 15/x, we can follow these steps:
Step 1: Rewrite the inequality without fractions.
Multiply both sides of the inequality by x to get rid of the fraction:
x(15x - 16) ≤ 15
Step 2: Simplify the inequality.
Distribute x to each term within the parentheses:
15x^2 - 16x ≤ 15
Step 3: Move all terms to one side of the inequality.
Subtract 15 from both sides of the inequality:
15x^2 - 16x - 15 ≤ 0
Step 4: Factor the quadratic expression on the left side.
This quadratic expression does not factor easily, so we can solve it using the quadratic formula or by graphing the equation and determining the solutions.
By using the quadratic formula, x is determined by:
x = (-b ± sqrt(b^2 - 4ac))/(2a)
In this case, a = 15, b = -16, and c = -15. Plugging these values into the quadratic formula, we get:
x = (-(-16) ± sqrt((-16)^2 - 4(15)(-15))) / (2*15)
Simplifying further:
x = (16 ± sqrt(256 + 900)) / 30
x = (16 ± sqrt(1156)) / 30
x = (16 ± 34) / 30
So, the solutions for x are:
x = (16 + 34) / 30 = 50 / 30 = 5/3
x = (16 - 34) / 30 = -18 / 30 = -3/5
Now, let's check the points x = 0, x = 5/3, and x = -3/5 to determine if they are included in the solution set:
1. x = 0:
Plugging x = 0 into the original inequality:
15(0) - 16 ≤ 15/0
-16 ≤ undefined
Since the right side is undefined, x = 0 is not included in the solution set.
2. x = 5/3:
Plugging x = 5/3 into the original inequality:
15(5/3) - 16 ≤ 15/(5/3)
25 - 16 ≤ 45/5
9 ≤ 9
Since 9 is indeed less than or equal to 9, x = 5/3 is included in the solution set.
3. x = -3/5:
Plugging x = -3/5 into the original inequality:
15(-3/5) - 16 ≤ 15/(-3/5)
-9 - 16 ≤ -25
-25 ≤ -25
Since -25 is indeed less than or equal to -25, x = -3/5 is included in the solution set.
Therefore, the solution set for the inequality 15x - 16 ≤ 15/x is: x ∈ {-3/5, 5/3}.