Solve for x under the assumption that x > 0. Enter your answer in interval notation using grouping symbols.
x - (10/x) < 9
I need someone to show me how to work through this type of problem, I don't understand it.
x - 10/x < 9
Since x>0, we can multiply by x and not change the direction of the inequality, so
x^2 - 10 < 9x
x^2 - 9x - 10 < 0
(x-10)(x+1) < 0
So, either
(A): (x-10)<0 and (x+1)>0
or
(B): (x-10)>0 and (x+1)<0
(A):
x<10 and x > -1 or -1 < x < 10
(B):
x > 10 and x < -1
can't happen
So, -1 < x < 10
But, we stipulated that x>0, so
0 < x < 10
x is in (0,10)
Or, if you remember what parabolas look like, you can note that since the parabola opens upward, (x-10)(x+1) < 0 when x is between the roots. -1 < x < 10
x-3/x<-2
To solve the inequality x - (10/x) < 9, we can follow these steps:
1. Move all terms to the left side of the inequality:
x - (10/x) - 9 < 0
2. Obtain a common denominator by multiplying both sides of the inequality by x:
x^2 - 10 - 9x < 0
3. Rearrange the terms in descending order:
x^2 - 9x - 10 < 0
4. Factorize the quadratic expression:
(x - 10)(x + 1) < 0
5. To find the values of x that make the inequality true, we need to consider the sign changes of the expression. The inequality will be satisfied when the expression is negative.
We create a sign chart to determine the intervals where the expression is negative:
|--------+--------+------|
-1 10 ∞
For x < -1, both factors are negative, so the expression is positive.
For -1 < x < 10, the first factor (x - 10) is negative, but the second factor (x + 1) is positive, so the expression is negative.
For x > 10, both factors are positive, so the expression is positive.
6. Using the sign chart, we see that the inequality is satisfied when -1 < x < 10.
Therefore, the solution in interval notation using grouping symbols is (-1, 10).
To solve the inequality x - (10/x) < 9 under the assumption that x > 0, follow these steps:
Step 1: Simplify the inequality:
- Begin by multiplying both sides of the inequality by x to eliminate the fraction:
(x - (10/x)) * x < 9 * x
- This simplifies to:
x^2 - 10 < 9x
Step 2: Rearrange the inequality in standard form:
- Move all terms to the left side:
x^2 - 9x - 10 < 0
Step 3: Factorize the quadratic expression:
- Look for two numbers that multiply to give -10 and add up to -9:
(x - 10)(x + 1) < 0
Step 4: Determine the critical points:
- We set each factor equal to zero and solve for x:
x - 10 = 0 => x = 10
x + 1 = 0 => x = -1
- These are the critical points which divide the number line into intervals.
Step 5: Construct the sign chart:
- Draw a number line and plot the critical points (-1 and 10) on it.
- We have two intervals: (-∞, -1) and (-1, 10).
Step 6: Test a value in each interval:
- Choose a value within each interval and substitute it into the inequality to determine the sign:
For the interval (-∞, -1), choose x = -2:
(x - 10)(x + 1) = (-2 - 10)(-2 + 1) = (-12)(-1) = 12 > 0
For the interval (-1, 10), choose x = 0:
(x - 10)(x + 1) = (0 - 10)(0 + 1) = (-10)(1) = -10 < 0
Step 7: Determine the solution using the sign chart:
- The inequality is satisfied when the product (x - 10)(x + 1) is less than zero (negative).
- From the sign chart, we see that the solution is x belonging to the interval (-1, 10).
Step 8: Write the solution in interval notation:
- In interval notation, the solution is (-1, 10) [using parentheses to indicate that -1 and 10 are not included in the interval].
Therefore, the solution to the inequality x - (10/x) < 9, under the assumption that x > 0, is (-1, 10) in interval notation.