Solve the inequality 2𝑥^2 − 3𝑥 + 1 < 0  and represent the solution on a number line.

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 \( 1 \).

---

### 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.

Learn how to solve the inequality 2x^2 - 3x + 1 < 0 using the quadratic formula and represent the solution on a number line. This problem covers key algebraic concepts, including quadratic equations, inequalities, and number line representations. Suitable for high school students.

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