Solve 54x^64 \geq 49x^59

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

x \in (- \infty, 0] \cup [\frac{49}{24}, + \infty)

Evaluate

54 x^{6} \times 4 \geq 49 x^{5} \times 9

Multiply the terms

216 x^{6} \geq 49 x^{5} \times 9

Multiply the terms

216 x^{6} \geq 441 x^{5}

Move the expression to the left side

216 x^{6} - 441 x^{5} \geq 0

Rewrite the expression

216 x^{6} - 441 x^{5} = 0

Factor the expression

9 x^{5} (24 x - 49) = 0

Divide both sides

x^{5} (24 x - 49) = 0

Separate the equation into 2 possible cases

x^{5} = 0 24 x - 49 = 0

The only way a power can be 0 is when the base equals 0

x = 0 24 x - 49 = 0

Solve the equation

More Steps

x = 0 x = \frac{49}{24}

Determine the test intervals using the critical values

x < 0 0 < x < \frac{49}{24} x > \frac{49}{24}

Choose a value form each interval

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

To determine if x < 0 is the solution to the inequality,test if the chosen value x = - 1 satisfies the initial inequality

More Steps

x < 0 is the solution x_{2} = 1 x_{3} = 3

To determine if 0 < x < \frac{49}{24} is the solution to the inequality,test if the chosen value x = 1 satisfies the initial inequality

More Steps

x < 0 is the solution 0 < x < \frac{49}{24} is not a solution x_{3} = 3

To determine if x > \frac{49}{24} is the solution to the inequality,test if the chosen value x = 3 satisfies the initial inequality

More Steps

x < 0 is the solution 0 < x < \frac{49}{24} is not a solution x > \frac{49}{24} is the solution

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

x \leq 0 is the solution x \geq \frac{49}{24} is the solution

Solution

x \in (- \infty, 0] \cup [\frac{49}{24}, + \infty)

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