Solve (7x 14)/(x-9)>= 2x/7

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

x \in (- \infty, 0] \cup (9, 352]

Evaluate

\frac{7 x \times 14}{x - 9} \geq 2 \times \frac{x}{7}

Find the domain

More Steps

\frac{7 x \times 14}{x - 9} \geq 2 \times \frac{x}{7}, x \neq = 9

Multiply the terms

\frac{98 x}{x - 9} \geq 2 \times \frac{x}{7}

Multiply the terms

\frac{98 x}{x - 9} \geq \frac{2 x}{7}

Move the expression to the left side

\frac{98 x}{x - 9} - \frac{2 x}{7} \geq 0

Subtract the terms

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\frac{704 x - 2 x ^{2}}{7 ( x - 9 )} \geq 0

Set the numerator and denominator of \frac{704 x - 2 x ^{2}}{7 ( x - 9 )} equal to 0 to find the values of x where sign changes may occur

704 x - 2 x^{2} = 0 7 (x - 9) = 0

Calculate

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x = 0 x = 352 7 (x - 9) = 0

Calculate

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x = 0 x = 352 x = 9

Determine the test intervals using the critical values

x < 0 0 < x < 9 9 < x < 352 x > 352

Choose a value form each interval

x_{1} = - 1 x_{2} = 5 x_{3} = 181 x_{4} = 353

To determine if x < 0 is the solution to the inequality,test if the chosen value x = - 1 satisfies the initial inequality

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x < 0 is the solution x_{2} = 5 x_{3} = 181 x_{4} = 353

To determine if 0 < x < 9 is the solution to the inequality,test if the chosen value x = 5 satisfies the initial inequality

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x < 0 is the solution 0 < x < 9 is not a solution x_{3} = 181 x_{4} = 353

To determine if 9 < x < 352 is the solution to the inequality,test if the chosen value x = 181 satisfies the initial inequality

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x < 0 is the solution 0 < x < 9 is not a solution 9 < x < 352 is the solution x_{4} = 353

To determine if x > 352 is the solution to the inequality,test if the chosen value x = 353 satisfies the initial inequality

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x < 0 is the solution 0 < x < 9 is not a solution 9 < x < 352 is the solution x > 352 is not a solution

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

x \leq 0 is the solution 9 < x \leq 352 is the solution

The final solution of the original inequality is x \in (- \infty, 0] \cup (9, 352]

x \in (- \infty, 0] \cup (9, 352]

Check if the solution is in the defined range

x \in (- \infty, 0] \cup (9, 352], x \neq = 9

Solution

x \in (- \infty, 0] \cup (9, 352]

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