To find the values of x that satisfy the inequality, we first need to find the critical points where the expression is equal to 0 and where it is undefined.
Setting the numerator (5x-4) equal to 0: 5x - 4 = 0 5x = 4 x = 4/5
Setting the denominator (7x(2x-1)) equal to 0: 7x(2x-1) = 0 7x = 0 or 2x - 1 = 0 x = 0 or x = 1/2
Now we create a number line with these critical points: 0, 1/2, and 4/5.
We test a value from each interval: For x < 0: Let x = -1, (-5-4) / (7(-14)) = -9 / -98 = 9/98 > 0 0 < x < 1/2: Let x = 1/4, (5/4-4) / (7(1/2)(1/2-1)) = (1/4) / (7(1/2)(-1/2)) = (1/4) / (7(1/4)) = 1/4 / 7/4 = 1/7 > 0 1/2 < x < 4/5: Let x = 3/4, (15/4-4) / (7(3/2)(2/3-1)) = (3/4) / (7(3/2)(-1/3)) = (3/4) / (7(1)) = 3/4 / 7 = 3/28 > 0 x > 4/5: Let x = 1, (5-4) / (72(2-1)) = 1 / (14) = 1/14 > 0
Therefore, the solution to the inequality is x < 0 or 0 < x < 1/2.
To find the values of x that satisfy the inequality, we first need to find the critical points where the expression is equal to 0 and where it is undefined.
Setting the numerator (5x-4) equal to 0:
5x - 4 = 0
5x = 4
x = 4/5
Setting the denominator (7x(2x-1)) equal to 0:
7x(2x-1) = 0
7x = 0 or 2x - 1 = 0
x = 0 or x = 1/2
Now we create a number line with these critical points: 0, 1/2, and 4/5.
We test a value from each interval:
For x < 0: Let x = -1, (-5-4) / (7(-14)) = -9 / -98 = 9/98 > 0
0 < x < 1/2: Let x = 1/4, (5/4-4) / (7(1/2)(1/2-1)) = (1/4) / (7(1/2)(-1/2)) = (1/4) / (7(1/4)) = 1/4 / 7/4 = 1/7 > 0
1/2 < x < 4/5: Let x = 3/4, (15/4-4) / (7(3/2)(2/3-1)) = (3/4) / (7(3/2)(-1/3)) = (3/4) / (7(1)) = 3/4 / 7 = 3/28 > 0
x > 4/5: Let x = 1, (5-4) / (72(2-1)) = 1 / (14) = 1/14 > 0
Therefore, the solution to the inequality is x < 0 or 0 < x < 1/2.