Solve the inequality:
x+8 over x-1 > OR = 0
Write the solution set in interval notation. Show work/explanation.
Here is how I do these...
First of all it should be obvious that x cannot be 1, or else we are dividing by zero.
I see 2 critical values, namely x = -8 and x = 1
this splits the number line into 3 sections,
1. less than -8
2. between -8 and 1
3. greater than 1
I then pick an "nice" number in each section and mentally check to see if it works in my inequation.
In most cases you don't actually have to work out the answer, just the correct sign of the answer.
1. let x= -10, .... is -/- ≥ 0 ? YES
2. let x = 0, .... is +/- ≥ 0 ? NO
3. let x = +10 ... is +/+ ≥ 0 ? YES
so x ≤ -8 OR x > 1 , x any real number.
You can change that into the "interval notation" that you were taught.
(3x2) (2x4)
To solve the inequality, we need to consider two cases: when the fraction is strictly greater than 0 and when it is equal to 0.
Case 1: x + 8/x - 1 > 0
To determine the sign of the fraction, we can look at the signs of the numerator and denominator.
When x > 1, both the numerator (x + 8) and the denominator (x - 1) are positive. Therefore, the fraction is positive.
When x < 1, both the numerator (x + 8) and the denominator (x - 1) are negative. In this case, dividing two negative numbers results in a positive fraction.
So, in order for the fraction x + 8/x - 1 to be positive, x must be either greater than 1 or less than 1.
Case 2: x + 8/x - 1 = 0
To solve for this case, we need to find the value of x that makes the fraction equal to zero.
Setting the numerator equal to zero:
x + 8 = 0
x = -8
Setting the denominator equal to zero:
x - 1 = 0
x = 1
So, the fraction is equal to 0 when x = -8 or x = 1.
In summary:
- If x > 1 or x < -8, the fraction x + 8/x - 1 is positive.
- If x = 1 or x = -8, the fraction x + 8/x - 1 is equal to 0.
Now, let's represent the solution set in interval notation:
For x > 1:
Solution: (1, +∞)
For x < -8:
Solution: (-∞, -8)
For x = 1 or x = -8:
Solution: {-8, 1}
Therefore, the overall solution set in interval notation is:
(-∞, -8) ∪ {-8, 1} ∪ (1, +∞)