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Answer:
1. Graphing Basics:
To plot the graph of the quadratic inequality x^2 - 4x - 12 < 0 on a number line, we need to find the critical points and shade the appropriate regions that satisfy the inequality.
First, let's find the critical points by solving the equation x^2 - 4x - 12 = 0. We can factorize or use the quadratic formula to find that the critical points are x = -2 and x = 6.
Now we can plot these critical points on the number line. Label -2 on the left side and 6 on the right side.
To determine which regions to shade, you can choose test points within each interval and substitute them into the inequality. For example, for the interval (-∞, -2), you can choose x = -3 as a test point. Substituting -3 into the inequality, we get (-3)^2 - 4(-3) - 12 < 0, which simplifies to 3 + 12 - 12 < 0. Since this is true, we shade the region to the left of -2.
Similarly, you can choose test points for the other intervals (-2, 6) and (6, ∞) to determine which regions to shade.
2. Comparing Inequalities:
To compare and contrast the number line representations of the inequalities x - 9 > 0 and x - 2x - 3 < 0, we need to analyze the shaded regions on the number line for each inequality.
For x - 9 > 0, we need to find the critical point by solving x - 9 = 0, which gives us x = 9. Since the inequality is greater than 0, we shade the region to the right of 9 on the number line.
For x - 2x - 3 < 0, we need to simplify the inequality to -x - 3 < 0, which further simplifies to -x < 3. Dividing both sides by -1 and reversing the inequality gives us x > -3. Since the inequality is less than 0, we shade the region to the left of -3 on the number line.
The difference in the shaded regions for each inequality is that x - 9 > 0 is shaded to the right of the critical point (9), while x - 2x - 3 < 0 is shaded to the left of the critical point (-3).
3. Identifying Critical Points:
For the quadratic inequality x^2 + 3x + 2 < 0, let's determine the critical points and explain their significance in terms of the graph on the number line.
To find the critical points, we solve the equation x^2 + 3x + 2 = 0. Factoring or using the quadratic formula, we find that the critical points are x = -1 and x = -2.
The significance of these critical points is that they represent the x-values where the quadratic curve intersects the x-axis. In other words, they are the solutions to the equation x^2 + 3x + 2 = 0, which are also known as the roots or x-intercepts of the quadratic equation.
On the number line, these critical points help us determine the intervals or regions where the inequality x^2 + 3x + 2 < 0 is satisfied. By testing values within these intervals, we can determine which regions to shade.
Please note that without access to a visual representation, it may be helpful to draw a simple number line and label the critical points and shaded regions to better understand the concept.