Solve 2x - 5 < 1/33x/4 (1-x)/3

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve for x

- \frac{7920 + 79 1 ^{2} + 791}{2} < x < \frac{7920 + 79 1 ^{2} - 791}{2}

Alternative Form

x \in (- \frac{7920 + 79 1 ^{2} + 791}{2}, \frac{7920 + 79 1 ^{2} - 791}{2})

Evaluate

2 x - 5 < \frac{1}{33} \times \frac{x}{4} \times \frac{1 - x}{3}

Multiply the terms

More Steps

2 x - 5 < \frac{x ( 1 - x )}{396}

Multiply both sides of the inequality by 396

(2 x - 5) \times 396 < \frac{x ( 1 - x )}{396} \times 396

Multiply the terms

More Steps

792 x - 1980 < \frac{x ( 1 - x )}{396} \times 396

Multiply the terms

792 x - 1980 < x (1 - x)

Move the expression to the left side

792 x - 1980 - x (1 - x) < 0

Subtract the terms

More Steps

791 x - 1980 + x^{2} < 0

Rewrite the expression

791 x - 1980 + x^{2} = 0

Add or subtract both sides

791 x + x^{2} = 1980

Add the same value to both sides

791 x + x^{2} + \frac{79 1 ^{2}}{4} = 1980 + \frac{79 1 ^{2}}{4}

Simplify the expression

(x + \frac{791}{2})^{2} = \frac{7920 + 79 1 ^{2}}{4}

Take the root of both sides of the equation and remember to use both positive and negative roots

x + \frac{791}{2} = \pm \frac{7920 + 79 1 ^{2}}{4}

Simplify the expression

x + \frac{791}{2} = \pm \frac{7920 + 79 1 ^{2}}{2}

Separate the equation into 2 possible cases

x + \frac{791}{2} = \frac{7920 + 79 1 ^{2}}{2} x + \frac{791}{2} = - \frac{7920 + 79 1 ^{2}}{2}

Solve the equation

More Steps

x = \frac{7920 + 79 1 ^{2} - 791}{2} x + \frac{791}{2} = - \frac{7920 + 79 1 ^{2}}{2}

Solve the equation

More Steps

x = \frac{7920 + 79 1 ^{2} - 791}{2} x = - \frac{7920 + 79 1 ^{2} + 791}{2}

Determine the test intervals using the critical values

x < - \frac{7920 + 79 1 ^{2} + 791}{2} - \frac{7920 + 79 1 ^{2} + 791}{2} < x < \frac{7920 + 79 1 ^{2} - 791}{2} x > \frac{7920 + 79 1 ^{2} - 791}{2}

Choose a value form each interval

x_{1} = - 794 x_{2} = - 396 x_{3} = 3

To determine if x < - \frac{7920 + 79 1 ^{2} + 791}{2} is the solution to the inequality,test if the chosen value x = - 794 satisfies the initial inequality

More Steps

x < - \frac{7920 + 79 1 ^{2} + 791}{2} is not a solution x_{2} = - 396 x_{3} = 3

To determine if - \frac{7920 + 79 1 ^{2} + 791}{2} < x < \frac{7920 + 79 1 ^{2} - 791}{2} is the solution to the inequality,test if the chosen value x = - 396 satisfies the initial inequality

More Steps

x < - \frac{7920 + 79 1 ^{2} + 791}{2} is not a solution - \frac{7920 + 79 1 ^{2} + 791}{2} < x < \frac{7920 + 79 1 ^{2} - 791}{2} is the solution x_{3} = 3

To determine if x > \frac{7920 + 79 1 ^{2} - 791}{2} is the solution to the inequality,test if the chosen value x = 3 satisfies the initial inequality

More Steps

x < - \frac{7920 + 79 1 ^{2} + 791}{2} is not a solution - \frac{7920 + 79 1 ^{2} + 791}{2} < x < \frac{7920 + 79 1 ^{2} - 791}{2} is the solution x > \frac{7920 + 79 1 ^{2} - 791}{2} is not a solution

Solution

- \frac{7920 + 79 1 ^{2} + 791}{2} < x < \frac{7920 + 79 1 ^{2} - 791}{2}

Alternative Form

x \in (- \frac{7920 + 79 1 ^{2} + 791}{2}, \frac{7920 + 79 1 ^{2} - 791}{2})

Show Solution