Solve log10(2x-2007/x 1) <= 0

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

x \in (- \infty, - \frac{3 446}{2}] \cup (0, \frac{3 446}{2}]

Evaluate

lo g_{10} (10) \times (2 x - \frac{2007}{x} \times 1) \leq 0

Find the domain

lo g_{10} (10) \times (2 x - \frac{2007}{x} \times 1) \leq 0, x \neq = 0

Simplify

More Steps

2 x - \frac{2007}{x} \leq 0

Rearrange the terms

\frac{2 x ^{2} - 2007}{x} \leq 0

Set the numerator and denominator of \frac{2 x ^{2} - 2007}{x} equal to 0 to find the values of x where sign changes may occur

2 x^{2} - 2007 = 0 x = 0

Calculate

More Steps

x = \frac{3 446}{2} x = - \frac{3 446}{2} x = 0

Determine the test intervals using the critical values

x < - \frac{3 446}{2} - \frac{3 446}{2} < x < 0 0 < x < \frac{3 446}{2} x > \frac{3 446}{2}

Choose a value form each interval

x_{1} = - 33 x_{2} = - 16 x_{3} = 16 x_{4} = 33

To determine if x < - \frac{3 446}{2} is the solution to the inequality,test if the chosen value x = - 33 satisfies the initial inequality

More Steps

x < - \frac{3 446}{2} is the solution x_{2} = - 16 x_{3} = 16 x_{4} = 33

To determine if - \frac{3 446}{2} < x < 0 is the solution to the inequality,test if the chosen value x = - 16 satisfies the initial inequality

More Steps

x < - \frac{3 446}{2} is the solution - \frac{3 446}{2} < x < 0 is not a solution x_{3} = 16 x_{4} = 33

To determine if 0 < x < \frac{3 446}{2} is the solution to the inequality,test if the chosen value x = 16 satisfies the initial inequality

More Steps

x < - \frac{3 446}{2} is the solution - \frac{3 446}{2} < x < 0 is not a solution 0 < x < \frac{3 446}{2} is the solution x_{4} = 33

To determine if x > \frac{3 446}{2} is the solution to the inequality,test if the chosen value x = 33 satisfies the initial inequality

More Steps

x < - \frac{3 446}{2} is the solution - \frac{3 446}{2} < x < 0 is not a solution 0 < x < \frac{3 446}{2} is the solution x > \frac{3 446}{2} is not a solution

The original inequality is a nonstrict inequality,so include the critical value in the solution

x \leq - \frac{3 446}{2} is the solution 0 < x \leq \frac{3 446}{2} is the solution

The final solution of the original inequality is x \in (- \infty, - \frac{3 446}{2}] \cup (0, \frac{3 446}{2}]

x \in (- \infty, - \frac{3 446}{2}] \cup (0, \frac{3 446}{2}]

Check if the solution is in the defined range

x \in (- \infty, - \frac{3 446}{2}] \cup (0, \frac{3 446}{2}], x \neq = 0

Solution

x \in (- \infty, - \frac{3 446}{2}] \cup (0, \frac{3 446}{2}]

Show Solution