https://file.aiquickdraw.com/mathbot/file/17282588765392a5njmzy.jpg

The inequality given in the image is:

\[
2x^2 - 6x + 4 < 5x - 8
\]

Let's solve it step-by-step.

### Step 1: Move all terms to one side of the inequality.
We want to bring all terms to the left-hand side. Subtract \( 5x \) and add 8 to both sides of the inequality:

\[
2x^2 - 6x + 4 - 5x + 8 < 0
\]

Simplifying:

\[
2x^2 - 11x + 12 < 0
\]

### Step 2: Solve the corresponding quadratic equation.
We first solve the quadratic equation \( 2x^2 - 11x + 12 = 0 \) to find the critical points.

Using the quadratic formula:

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

Here, \( a = 2 \), \( b = -11 \), and \( c = 12 \). Substituting these into the formula:

\[
x = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(2)(12)}}{2(2)}
\]

\[
x = \frac{11 \pm \sqrt{121 - 96}}{4}
\]

\[
x = \frac{11 \pm \sqrt{25}}{4}
\]

\[
x = \frac{11 \pm 5}{4}
\]

So, the solutions are:

\[
x_1 = \frac{11 + 5}{4} = \frac{16}{4} = 4
\]
\[
x_2 = \frac{11 - 5}{4} = \frac{6}{4} = \frac{3}{2}
\]

### Step 3: Analyze the inequality.
The quadratic inequality \( 2x^2 - 11x + 12 < 0 \) is satisfied between the roots \( x_1 = 4 \) and \( x_2 = \frac{3}{2} \).

Since the parabola opens upwards (because the coefficient of \( x^2 \) is positive), the inequality is true between the roots:

\[
\frac{3}{2} < x < 4
\]

### Step 4: Write the solution in interval notation.
The solution in interval notation is:

\[
\left( \frac{3}{2}, 4 \right)
\]

---

Would you like any further details or clarifications?

Here are 5 related questions you might find interesting:
1. How do you solve inequalities that include absolute values?
2. Can inequalities have no solutions or infinite solutions? If so, how do you identify them?
3. How can you check the solution of a quadratic inequality graphically?
4. What happens if the quadratic inequality has no real roots?
5. How do we deal with quadratic inequalities when the leading coefficient is negative?

**Tip:** Always make sure to reverse the inequality symbol when multiplying or dividing both sides by a negative number.

Solve the following inequality. Give your answer in interval notation. 2x^2 - 6x + 4 < 5x - 8

This solution explains how to solve the quadratic inequality 2x^2 - 6x + 4 < 5x - 8 step-by-step, including moving terms, using the quadratic formula, and finding the solution in interval notation. Suitable for high school students in grades 10-12.

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