Solve 3x^2 >= 5x-2

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

Solve for x

x \in (- \infty, \frac{2}{3}] \cup [1, + \infty)

Evaluate

3 x^{2} \geq 5 x - 2

Move the expression to the left side

3 x^{2} - (5 x - 2) \geq 0

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

3 x^{2} - 5 x + 2 \geq 0

Rewrite the expression

3 x^{2} - 5 x + 2 = 0

Factor the expression

More Steps

(x - 1) (3 x - 2) = 0

When the product of factors equals 0,at least one factor is 0

x - 1 = 0 3 x - 2 = 0

Solve the equation for x

More Steps

x = 1 3 x - 2 = 0

Solve the equation for x

More Steps

x = 1 x = \frac{2}{3}

Determine the test intervals using the critical values

x < \frac{2}{3} \frac{2}{3} < x < 1 x > 1

Choose a value form each interval

x_{1} = - 1 x_{2} = \frac{5}{6} x_{3} = 2

To determine if x < \frac{2}{3} is the solution to the inequality,test if the chosen value x = - 1 satisfies the initial inequality

More Steps

x < \frac{2}{3} is the solution x_{2} = \frac{5}{6} x_{3} = 2

To determine if \frac{2}{3} < x < 1 is the solution to the inequality,test if the chosen value x = \frac{5}{6} satisfies the initial inequality

More Steps

x < \frac{2}{3} is the solution \frac{2}{3} < x < 1 is not a solution x_{3} = 2

To determine if x > 1 is the solution to the inequality,test if the chosen value x = 2 satisfies the initial inequality

More Steps

x < \frac{2}{3} is the solution \frac{2}{3} < x < 1 is not a solution x > 1 is the solution

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

x \leq \frac{2}{3} is the solution x \geq 1 is the solution

Solution

x \in (- \infty, \frac{2}{3}] \cup [1, + \infty)

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