To solve the inequality $2x^2 - 3x + 1 < 0$, follow these steps:

Step 1: Find the roots of the corresponding quadratic equation

The quadratic equation is: $2x^2 - 3x + 1 = 0$

To find the roots, use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 2$, $b = -3$, and $c = 1$.

Substituting the values, we get: $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(1)}}{2(2)}$ $x = \frac{3 \pm \sqrt{9 - 8}}{4}$ $x = \frac{3 \pm \sqrt{1}}{4}$ $x = \frac{3 \pm 1}{4}$

So, the roots are: $x = \frac{4}{4} = 1 \quad \text{and} \quad x = \frac{2}{4} = \frac{1}{2}$

Step 2: Analyze the sign of the quadratic expression

The roots divide the number line into three intervals:

• $x < \frac{1}{2}$
• $\frac{1}{2} < x < 1$
• $x > 1$

Determine the sign of $2x^2 - 3x + 1$ in each interval by testing a point from each interval:

1. For $x < \frac{1}{2}$ (e.g., $x = 0$): $2(0)^2 - 3(0) + 1 = 1 \quad (\text{positive})$

2. For $\frac{1}{2} < x < 1$ (e.g., $x = \frac{3}{4}$): $2\left(\frac{3}{4}\right)^2 - 3\left(\frac{3}{4}\right) + 1 = 2\left(\frac{9}{16}\right) - \frac{9}{4} + 1 = \frac{9}{8} - \frac{9}{4} + 1 = \frac{9}{8} - \frac{18}{8} + \frac{8}{8} = \frac{-1}{8} \quad (\text{negative})$

3. For $x > 1$ (e.g., $x = 2$): $2(2)^2 - 3(2) + 1 = 8 - 6 + 1 = 3 \quad (\text{positive})$

Step 3: Write the solution set

The quadratic expression $2x^2 - 3x + 1$ is negative between $\frac{1}{2}$ and $1$, so the solution to the inequality is: $\frac{1}{2} < x < 1$

Step 4: Represent the solution on a number line

On the number line, the interval $\left(\frac{1}{2}, 1\right)$ is represented as follows:

• Draw a number line.
• Mark $\frac{1}{2}$ and $1$ on the number line.
• Draw an open circle at both $\frac{1}{2}$ and $1$ (since the inequality is strict, "<").
• Shade the region between $\frac{1}{2}$ and $1$.

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Do you want any more details or have any questions?

Here are five relative questions to expand on this topic:

1. How would the solution change if the inequality were $2x^2 - 3x + 1 \leq 0$?
2. What is the graphical representation of the function $y = 2x^2 - 3x + 1$?
3. How do you solve a quadratic inequality when the discriminant is negative?
4. What does it mean when the quadratic expression has no real roots?
5. How can you use the vertex form of a quadratic function to solve inequalities?

Tip: Always check the sign of the quadratic expression in each interval to correctly determine where it is positive or negative.