Solve (8x - 9)/5 >= (x ^ 2)/3

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

Solve for x

\frac{9}{5} \leq x \leq 3

Alternative Form

x \in [\frac{9}{5}, 3]

Evaluate

\frac{8 x - 9}{5} \geq \frac{x ^{2}}{3}

Multiply both sides of the inequality by 15

\frac{8 x - 9}{5} \times 15 \geq \frac{x ^{2}}{3} \times 15

Multiply the terms

More Steps

24 x - 27 \geq \frac{x ^{2}}{3} \times 15

Multiply the terms

More Steps

24 x - 27 \geq 5 x^{2}

Move the expression to the left side

24 x - 27 - 5 x^{2} \geq 0

Rewrite the expression

24 x - 27 - 5 x^{2} = 0

Factor the expression

More Steps

(- 3 + x) (9 - 5 x) = 0

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

- 3 + x = 0 9 - 5 x = 0

Solve the equation for x

More Steps

x = 3 9 - 5 x = 0

Solve the equation for x

More Steps

x = 3 x = \frac{9}{5}

Determine the test intervals using the critical values

x < \frac{9}{5} \frac{9}{5} < x < 3 x > 3

Choose a value form each interval

x_{1} = 1 x_{2} = 2 x_{3} = 4

To determine if x < \frac{9}{5} is the solution to the inequality,test if the chosen value x = 1 satisfies the initial inequality

More Steps

x < \frac{9}{5} is not a solution x_{2} = 2 x_{3} = 4

To determine if \frac{9}{5} < x < 3 is the solution to the inequality,test if the chosen value x = 2 satisfies the initial inequality

More Steps

x < \frac{9}{5} is not a solution \frac{9}{5} < x < 3 is the solution x_{3} = 4

To determine if x > 3 is the solution to the inequality,test if the chosen value x = 4 satisfies the initial inequality

More Steps

x < \frac{9}{5} is not a solution \frac{9}{5} < x < 3 is the solution x > 3 is not a solution

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

\frac{9}{5} \leq x \leq 3 is the solution

Solution

\frac{9}{5} \leq x \leq 3

Alternative Form

x \in [\frac{9}{5}, 3]

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