Question

Question was provided via image upload.

Explanation:

Identify the inequality

The blurry text at the top left of the image shows a quadratic inequality. By analyzing the visible characters and the number line marked at \(x = 4\), the inequality is:

$$x(x - 3) > 4$$

We can rewrite this inequality in standard form:

$$x^2 - 3x - 4 > 0$$

Find the boundary points

To find the critical boundary points, we solve the corresponding quadratic equation:

$$x^2 - 3x - 4 = 0$$

Factoring the quadratic expression:

$$(x - 4)(x + 1) = 0$$

This gives the boundary points:

$$x = -1 \quad \text{and} \quad x = 4$$

Determine the solution intervals

The inequality is \( (x - 4)(x + 1) > 0 \). We test the intervals defined by the boundary points \(x = -1\) and \(x = 4\):

• For \(x < -1\) (e.g., \(x = -2\)): \((-2 - 4)(-2 + 1) = (-6)(-1) = 6 > 0\) (True)
• For \(-1 < x < 4\) (e.g., \(x = 0\)): \((0 - 4)(0 + 1) = -4 > 0\) (False)
• For \(x > 4\) (e.g., \(x = 5\)): \((5 - 4)(5 + 1) = (1)(6) = 6 > 0\) (True)

Thus, the solution set is:

$$x < -1 \quad \text{or} \quad x > 4$$

Represent on the number line

Since the inequality uses a strict inequality sign (\(>\)), the endpoints at \(x = -1\) and \(x = 4\) must be open circles. The shaded regions represent values less than \(-1\) and values greater than \(4\), which are represented by two rays pointing outwards:

• An open circle at \(-1\) with a ray pointing to the left.
• An open circle at \(4\) with a ray pointing to the right.

Answer:

To solve and graph the inequality \(x(x - 3) > 4\):

1. Rewrite the inequality in standard form:

$$x^2 - 3x - 4 > 0$$

1. Factor the quadratic expression to find the critical boundary points:

$$(x - 4)(x + 1) > 0$$

The boundary points are <blank>\(x = -1\)</blank> and <blank>\(x = 4\)</blank>.

1. Test the intervals to find where the inequality holds true:

• The solution is <blank>\(x < -1\)</blank> or <blank>\(x > 4\)</blank>.

1. Graph the solution on the number line:

• Place an <blank>open</blank> circle at \(-1\) and draw a ray pointing to the left.
• Place an <blank>open</blank> circle at \(4\) and draw a ray pointing to the right.