Question

5x2 – 12x + 4 > 0; ​

Answer 1

5x2 – 12x + 4 > 0; ​

10 - 12x + 4 > 0

14 - 12x > 0

- 12x > - 14

x < 7/6

Answer 2

To solve the inequality $5x^2 - 12x + 4 > 0$, we can follow these steps:

Step 1: Find the critical points by setting the expression equal to zero and solving for $x$. Step 2: Analyze the sign of the expression in the intervals created by the critical points. Step 3: Determine the solution by identifying the intervals where the expression is greater than zero.

Let's go through each step:

Step 1: Find the critical points To find the critical points, we set the expression equal to zero:

$5x^2 - 12x + 4 = 0$

We can solve this quadratic equation using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

where $a = 5$, $b = -12$, and $c = 4$.

$x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \cdot 5 \cdot 4}}{2 \cdot 5}$

$x = \frac{12 \pm \sqrt{144 - 80}}{10}$

$x = \frac{12 \pm \sqrt{64}}{10}$

$x = \frac{12 \pm 8}{10}$

Now, we get two critical points:

$x_1 = \frac{12 + 8}{10} = \frac{20}{10} = 2$

$x_2 = \frac{12 - 8}{10} = \frac{4}{10} = 0.4$

Step 2: Analyze the sign of the expression in the intervals Now that we have the critical points, we can analyze the sign of the expression in the intervals created by these points. We choose test points from each interval and plug them into the inequality $5x^2 - 12x + 4 > 0$.

Test point 1: $x = 0$ (interval: $-\infty$ to $0.4$) $5(0)^2 - 12(0) + 4 = 4$, which is greater than zero.

Test point 2: $x = 1$ (interval: $0.4$ to $2$) $5(1)^2 - 12(1) + 4 = -3$, which is not greater than zero.

Test point 3: $x = 3$ (interval: $2$ to $+\infty$) $5(3)^2 - 12(3) + 4 = 19$, which is greater than zero.

Step 3: Determine the solution The solution to the inequality is the interval where the expression is greater than zero. From the analysis of the test points, we find that the inequality is satisfied for $x$ in the intervals $(-\infty, 0.4)$ and $(2, +\infty)$.

Thus, the solution to the inequality $5x^2 - 12x + 4 > 0$ is:

$x < 0.4$ or $x > 2$