Solve 2-(x-3)(x 1) >= -4

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve for x

\frac{- 33 + 3}{2} \leq x \leq \frac{33 + 3}{2}

Alternative Form

x \in [\frac{- 33 + 3}{2}, \frac{33 + 3}{2}]

Evaluate

2 - (x - 3) (x \times 1) \geq - 4

Remove the parentheses

2 - (x - 3) x \times 1 \geq - 4

Multiply the terms

More Steps

2 - x (x - 3) \geq - 4

Move the expression to the left side

2 - x (x - 3) - (- 4) \geq 0

Subtract the terms

More Steps

6 - x^{2} + 3 x \geq 0

Rewrite the expression

6 - x^{2} + 3 x = 0

Add or subtract both sides

- x^{2} + 3 x = - 6

Divide both sides

\frac{- x ^{2} + 3 x}{- 1} = \frac{- 6}{- 1}

Evaluate

x^{2} - 3 x = 6

Add the same value to both sides

x^{2} - 3 x + \frac{9}{4} = 6 + \frac{9}{4}

Simplify the expression

(x - \frac{3}{2})^{2} = \frac{33}{4}

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

x - \frac{3}{2} = \pm \frac{33}{4}

Simplify the expression

x - \frac{3}{2} = \pm \frac{33}{2}

Separate the equation into 2 possible cases

x - \frac{3}{2} = \frac{33}{2} x - \frac{3}{2} = - \frac{33}{2}

Solve the equation

More Steps

x = \frac{33 + 3}{2} x - \frac{3}{2} = - \frac{33}{2}

Solve the equation

More Steps

x = \frac{33 + 3}{2} x = \frac{- 33 + 3}{2}

Determine the test intervals using the critical values

x < \frac{- 33 + 3}{2} \frac{- 33 + 3}{2} < x < \frac{33 + 3}{2} x > \frac{33 + 3}{2}

Choose a value form each interval

x_{1} = - 2 x_{2} = 2 x_{3} = 5

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

More Steps

x < \frac{- 33 + 3}{2} is not a solution x_{2} = 2 x_{3} = 5

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

More Steps

x < \frac{- 33 + 3}{2} is not a solution \frac{- 33 + 3}{2} < x < \frac{33 + 3}{2} is the solution x_{3} = 5

To determine if x > \frac{33 + 3}{2} is the solution to the inequality,test if the chosen value x = 5 satisfies the initial inequality

More Steps

x < \frac{- 33 + 3}{2} is not a solution \frac{- 33 + 3}{2} < x < \frac{33 + 3}{2} is the solution x > \frac{33 + 3}{2} is not a solution

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

\frac{- 33 + 3}{2} \leq x \leq \frac{33 + 3}{2} is the solution

Solution

\frac{- 33 + 3}{2} \leq x \leq \frac{33 + 3}{2}

Alternative Form

x \in [\frac{- 33 + 3}{2}, \frac{33 + 3}{2}]

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