Solve x: (x - 11) - 5 < (2x - 11) 1

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

x \in (\frac{- 313 + 29}{4}, 11) \cup (\frac{313 + 29}{4}, + \infty)

Evaluate

x \div (x - 11) - 5 < (2 x - 11) \times 1

Find the domain

More Steps

x \div (x - 11) - 5 < (2 x - 11) \times 1, x \neq = 11

Rewrite the expression

\frac{x}{x - 11} - 5 < (2 x - 11) \times 1

Any expression multiplied by 1 remains the same

\frac{x}{x - 11} - 5 < 2 x - 11

Move the expression to the left side

\frac{x}{x - 11} - 5 - (2 x - 11) < 0

Subtract the terms

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\frac{29 x - 66 - 2 x ^{2}}{x - 11} < 0

Set the numerator and denominator of \frac{29 x - 66 - 2 x ^{2}}{x - 11} equal to 0 to find the values of x where sign changes may occur

29 x - 66 - 2 x^{2} = 0 x - 11 = 0

Calculate

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x = \frac{313 + 29}{4} x = \frac{- 313 + 29}{4} x - 11 = 0

Calculate

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x = \frac{313 + 29}{4} x = \frac{- 313 + 29}{4} x = 11

Determine the test intervals using the critical values

x < \frac{- 313 + 29}{4} \frac{- 313 + 29}{4} < x < 11 11 < x < \frac{313 + 29}{4} x > \frac{313 + 29}{4}

Choose a value form each interval

x_{1} = 2 x_{2} = 7 x_{3} = \frac{313 + 73}{8} x_{4} = 13

To determine if x < \frac{- 313 + 29}{4} is the solution to the inequality,test if the chosen value x = 2 satisfies the initial inequality

More Steps

x < \frac{- 313 + 29}{4} is not a solution x_{2} = 7 x_{3} = \frac{313 + 73}{8} x_{4} = 13

To determine if \frac{- 313 + 29}{4} < x < 11 is the solution to the inequality,test if the chosen value x = 7 satisfies the initial inequality

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x < \frac{- 313 + 29}{4} is not a solution \frac{- 313 + 29}{4} < x < 11 is the solution x_{3} = \frac{313 + 73}{8} x_{4} = 13

To determine if 11 < x < \frac{313 + 29}{4} is the solution to the inequality,test if the chosen value x = \frac{313 + 73}{8} satisfies the initial inequality

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x < \frac{- 313 + 29}{4} is not a solution \frac{- 313 + 29}{4} < x < 11 is the solution 11 < x < \frac{313 + 29}{4} is not a solution x_{4} = 13

To determine if x > \frac{313 + 29}{4} is the solution to the inequality,test if the chosen value x = 13 satisfies the initial inequality

More Steps

x < \frac{- 313 + 29}{4} is not a solution \frac{- 313 + 29}{4} < x < 11 is the solution 11 < x < \frac{313 + 29}{4} is not a solution x > \frac{313 + 29}{4} is the solution

The original inequality is a strict inequality,so does not include the critical value ,the final solution is x \in (\frac{- 313 + 29}{4}, 11) \cup (\frac{313 + 29}{4}, + \infty)

x \in (\frac{- 313 + 29}{4}, 11) \cup (\frac{313 + 29}{4}, + \infty)

Check if the solution is in the defined range

x \in (\frac{- 313 + 29}{4}, 11) \cup (\frac{313 + 29}{4}, + \infty), x \neq = 11

Solution

x \in (\frac{- 313 + 29}{4}, 11) \cup (\frac{313 + 29}{4}, + \infty)

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