x/4x+1 < 5x/6 solve the inequality answer in interval notation and show work - thanks
I will assume you meant
x/(4x+1) < 5x/6
times 6(4x+1)
6x < 20x^2 + 5x
20x^2 - x > 0
x(20x - 1) > 0
critical values: x = 0, x = 1/20 and of course x ≠ -1/4
try values in the intervals
1. x < -1/4, say x = -1
is -1/-3 < -5/6 ? , is 1/3 < -5/6 ? NO
2. between -1/4 and 0, say x = -.2
is -.2/.2 < -1/6 ? , is -1 < - 1/6 ? YES
3. between 0 and 1/20, say x = .01
is .01/1.04 < .05/6 ? , is .0096.. < .00833 ? , NO
4. greater than 1/20, say x = 1
is 1/5 < 5/6 ? Yes
So -1/4 < x < 0 OR x > 1/20
check my arithmetic
Poswt it.
To solve the inequality (x/4x + 1) < (5x/6), we will follow these steps and show the work involved.
Step 1: Simplify both sides of the inequality.
Start by simplifying the left side of the inequality:
(x/4x + 1) = (1/4 + 1) = (1 + 4)/(4x) = 5/(4x)
The inequality now becomes:
(5/(4x)) < (5x/6)
Step 2: Multiply both sides of the inequality by the common denominator to eliminate the fractions.
In this case, the common denominator is 24x.
Multiply both sides of the inequality by 24x:
24x * (5/(4x)) < 24x * (5x/6)
Simplifying further:
120 < 20x^2
Step 3: Rearrange the inequality and set it equal to zero.
To do this, subtract 20x^2 from both sides:
20x^2 - 120 > 0
Step 4: Factor the quadratic expression and solve for x.
Divide the inequality by 20 to simplify:
x^2 - 6 > 0
Now, factor the quadratic expression:
(x - √6)(x + √6) > 0
Step 5: Determine the sign of the expression for different intervals of x.
To solve the inequality, we will look at the sign of the expression (x - √6)(x + √6) for different intervals of x.
For x < -√6:
Both factors (x - √6) and (x + √6) are negative, hence the expression is positive.
For -√6 < x < √6:
The factor (x - √6) is negative, but (x + √6) is positive, resulting in a negative expression.
For x > √6:
Both factors (x - √6) and (x + √6) are positive, making the expression positive.
Step 6: Determine the solution in interval notation.
Since the original inequality is "greater than" (>) rather than "greater than or equal to" (≥), we exclude the interval where the expression is equal to zero.
The solution in interval notation is: (-∞, -√6) U (√6, +∞)
So, the solution to the inequality (x/4x + 1) < (5x/6) is (-∞, -√6) U (√6, +∞).