solve the following inequality
70x-51<70/x
I get this--but I do not know how to write the final answer in a solution set
70x-51<70/x
70x^2/x - 51x /x - 70/x < 0
70x^2 - 51x - 70/x <0
(10x + 7) (7x - 10)/x <0
x=0
(10x + 7)= 0
(7x - 10)=0
Can you please help me to write the answer in a solution set???
when you get to this point:
70x^2/x - 51x /x - 70/x < 0
multiply by x to get
70x^2 - 51x - 70 < 0 and note that x≠0
since 1/x is not defined for x=0
now you have
(10x+7)(7x-10) < 0
At this point you should have some idea what you are looking for. You know it is a parabola which opens upward, and crosses the x-axis at -7/10 and 10/7.
So, the interval between the roots satisfies the original inequality:
-7/10 < x < 10/7
Algebraically, since
(10x+7)(7x-10) < 0, either
(10x+7) < 0 and (7x-10) > 0
or
(10x+7) > 0 and (7x-10) < 0
solve those and you will find a solution set which agrees to the interval above.
oops. because x cannot be 0, the final solution is
(-7/10,0) U (0,10/7)
To solve the inequality (10x + 7)(7x - 10)/x < 0, we first find the critical points by setting each factor equal to zero:
10x + 7 = 0 => x = -7/10
7x - 10 = 0 => x = 10/7
x = 0
Next, we create a number line with these critical points:
-∞ |----- -7/10 -----|----- 0 -----|----- 10/7 -----|----- +∞
Now, we choose test points in each of the intervals created by the critical points and evaluate the inequality to find the sign of the expression. For simplicity, we can choose -1, 1, and 2 as the test points.
For x = -1:
(10(-1) + 7)(7(-1) - 10)/(-1) = -3(-17)/(-1) = 51
For x = 1:
(10(1) + 7)(7(1) - 10)/(1) = 17(-3)/(1) = -51
For x = 2:
(10(2) + 7)(7(2) - 10)/(2) = 37(4)/(2) = 74
From these evaluations, we can determine the sign of the expression in each interval:
-∞ |----- -7/10 ----- (-51) ----- [0] ----- 74 -----|----- 10/7 -----|----- +∞
Since the inequality is less than zero (<0), we are looking for the intervals where the expression is negative.
Thus, the solution to the inequality (10x + 7)(7x - 10)/x < 0 is:
(-∞, -7/10) ∪ (0, 10/7)