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The problem in the image requires finding the solution to the inequality \( 4x^2 + 7x - 2 \geq 0 \) and then expressing the solution in both Set-Builder Notation and Interval Notation.

### Step 1: Solve the inequality \( 4x^2 + 7x - 2 \geq 0 \)
We first need to solve the corresponding quadratic equation:
\[
4x^2 + 7x - 2 = 0
\]
We can solve this quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 4 \), \( b = 7 \), and \( c = -2 \).

\[
x = \frac{-7 \pm \sqrt{7^2 - 4(4)(-2)}}{2(4)}
\]
\[
x = \frac{-7 \pm \sqrt{49 + 32}}{8}
\]
\[
x = \frac{-7 \pm \sqrt{81}}{8}
\]
\[
x = \frac{-7 \pm 9}{8}
\]
So, we have two solutions:
\[
x_1 = \frac{-7 + 9}{8} = \frac{2}{8} = \frac{1}{4}
\]
\[
x_2 = \frac{-7 - 9}{8} = \frac{-16}{8} = -2
\]

### Step 2: Determine the sign of \( 4x^2 + 7x - 2 \)
The quadratic opens upwards because the coefficient of \( x^2 \) is positive. The roots divide the number line into three intervals:
1. \( (-\infty, -2) \)
2. \( (-2, \frac{1}{4}) \)
3. \( (\frac{1}{4}, \infty) \)

We test the sign of the quadratic expression in each interval:
- For \( x \in (-\infty, -2) \), pick \( x = -3 \):
  \[
  4(-3)^2 + 7(-3) - 2 = 36 - 21 - 2 = 13 > 0
  \]
- For \( x \in (-2, \frac{1}{4}) \), pick \( x = 0 \):
  \[
  4(0)^2 + 7(0) - 2 = -2 < 0
  \]
- For \( x \in (\frac{1}{4}, \infty) \), pick \( x = 1 \):
  \[
  4(1)^2 + 7(1) - 2 = 4 + 7 - 2 = 9 > 0
  \]

So, the quadratic expression is positive (or zero) on the intervals \( (-\infty, -2] \) and \( [\frac{1}{4}, \infty) \).

### Step 3: Write the solution in Set-Builder and Interval Notation
- **Set-Builder Notation:**
  \[
  \{ x \,|\, x \leq -2 \text{ or } x \geq \frac{1}{4} \}
  \]

- **Interval Notation:**
  \[
  (-\infty, -2] \cup \left[\frac{1}{4}, \infty\right)
  \]

Would you like more details on any step? Here are some related questions:

1. How do you determine the sign of a quadratic expression in different intervals?
2. What is the significance of the direction in which a parabola opens?
3. How does the quadratic formula work, and why is it applicable here?
4. Can we solve the quadratic inequality using a different method, like factoring?
5. What is the difference between Set-Builder Notation and Interval Notation?

**Tip:** When solving inequalities, always consider the intervals between the roots and test them to understand the behavior of the quadratic function.

Find the solution of the inequality 4x^2 + 7x - 2 ≥ 0 and express the solution in both Set-Builder Notation and Interval Notation.

This guide provides a detailed step-by-step solution to the quadratic inequality 4x^2 + 7x - 2 ≥ 0, including the use of the quadratic formula, sign analysis, and the expression of the solution in both Set-Builder and Interval Notation. Ideal for students in Grades 9-11.

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