Solve x/2 (x-2)/3 <5

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve for x

- 31 + 1 < x < 31 + 1

Alternative Form

x \in (- 31 + 1, 31 + 1)

Evaluate

\frac{x}{2} \times \frac{x - 2}{3} < 5

Multiply the terms

More Steps

\frac{x ( x - 2 )}{6} < 5

Multiply both sides of the inequality by 6

\frac{x ( x - 2 )}{6} \times 6 < 5 \times 6

Multiply the terms

x (x - 2) < 5 \times 6

Multiply the terms

x (x - 2) < 30

Move the expression to the left side

x (x - 2) - 30 < 0

Expand the expression

More Steps

x^{2} - 2 x - 30 < 0

Rewrite the expression

x^{2} - 2 x - 30 = 0

Add or subtract both sides

x^{2} - 2 x = 30

Add the same value to both sides

x^{2} - 2 x + 1 = 30 + 1

Simplify the expression

(x - 1)^{2} = 31

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

x - 1 = \pm 31

Separate the equation into 2 possible cases

x - 1 = 31 x - 1 = - 31

Move the constant to the right-hand side and change its sign

x = 31 + 1 x - 1 = - 31

Move the constant to the right-hand side and change its sign

x = 31 + 1 x = - 31 + 1

Determine the test intervals using the critical values

x < - 31 + 1 - 31 + 1 < x < 31 + 1 x > 31 + 1

Choose a value form each interval

x_{1} = - 6 x_{2} = 1 x_{3} = 8

To determine if x < - 31 + 1 is the solution to the inequality,test if the chosen value x = - 6 satisfies the initial inequality

More Steps

x < - 31 + 1 is not a solution x_{2} = 1 x_{3} = 8

To determine if - 31 + 1 < x < 31 + 1 is the solution to the inequality,test if the chosen value x = 1 satisfies the initial inequality

More Steps

x < - 31 + 1 is not a solution - 31 + 1 < x < 31 + 1 is the solution x_{3} = 8

To determine if x > 31 + 1 is the solution to the inequality,test if the chosen value x = 8 satisfies the initial inequality

More Steps

x < - 31 + 1 is not a solution - 31 + 1 < x < 31 + 1 is the solution x > 31 + 1 is not a solution

Solution

- 31 + 1 < x < 31 + 1

Alternative Form

x \in (- 31 + 1, 31 + 1)

Show Solution