Question

Solve (using any method), write your answer in interval notation and graph the solution set.

1a. x^2-6> -5x

1b. 6x^2-5x+1 less than or equal to 0

Answer 1

Conditions

Solve (using any method), write your answer in interval notation and graph the solution set.

1a. $x^2 - 6$ - 5x

1b. $6x^2 - 5x + 1$ less than or equal to 0

Solution

1a.

$$x^2 - 6 > -5x$$ $$x^2 + 5x - 6 > 0$$ $$x^2 + 5x - 6 = 0$$ $$D = 25 + 24 = 49$$ $$x = \frac{-5 \pm 7}{2}$$ $$x_1 = -6$$ $$x_2 = 1$$ $$(x - 1)(x + 6) > 0$$ $$x \in (-\infty, -6) \cup (1, +\infty)$$

1b.

$$6x^2 - 5x + 1 \leq 0$$ $$6x^2 - 5x + 1 = 0$$ $$\begin{array}{l} D = 25 - 24 = 1 \\ x = \frac{5 \pm 1}{12} \\ x_1 = \frac{1}{2} \\ x_2 = \frac{1}{3} \\ \left(x - \frac{1}{2}\right) \left(x - \frac{1}{3}\right) \leq 0 \\ x \in \left[\frac{1}{3}, \frac{1}{2}\right] \\ \end{array}$$