Problem 51 - Algebra 2 and Trigonometry

Question

In \(44-51 :\) a. Graph the given inequality. b. Determine if the given point is in the solution set. $$ 6 x^{2}+x \leq 2-y ;(10,0) $$

Step-by-Step Solution

Answer

The point \((10, 0)\) is not in the solution set of the inequality.

Step 1: Convert Inequality

Start by rearranging the given inequality into a standard form that can be easily graphed. The inequality is originally given as: \[ 6x^2 + x \leq 2 - y \] Convert it to the form suitable for graphing by solving for \( y \): \[ y \leq 2 - 6x^2 - x \] This can be considered as the equation of a parabola, \( y = 2 - 6x^2 - x \), with a boundary represented by the inequality \( \leq \).

Step 2: Graphing the Parabola

Graph the equation \( y = 2 - 6x^2 - x \). This is a downward-opening parabola because the term \(-6x^2\) has a negative coefficient. To graph, find the y-intercept by setting \( x = 0 \): \[ y = 2 - 6(0)^2 - 0 = 2 \] Thus, the y-intercept is at \( (0, 2) \). Now, determine other points and sketch the parabola.

Step 3: Shade the Solution Region

Since the inequality is \( y \leq 2 - 6x^2 - x \), the solution area is below or on the parabola. After sketching the parabola, shade the region beneath it to represent the set of points that satisfy the inequality.

Step 4: Test the Given Point

Check the given point \((10, 0)\) to see if it lies in the shaded region. Substitute \( x = 10 \) and \( y = 0 \) into the equation derived from the inequality: \[ 0 \leq 2 - 6(10)^2 - 10 \] Calculate: \[ 0 \leq 2 - 600 - 10 \] \[ 0 \leq -608 \] This statement is false, indicating that the point \((10, 0)\) does not satisfy the inequality.

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions where the highest exponent of the variable is 2. In this exercise, we start with the inequality \(6x^2 + x \leq 2 - y\). This type of inequality can be somewhat tricky because it doesn't represent a single line or curve but rather a region of the graph where the inequality holds true.
To solve a quadratic inequality:

• We often convert it to a standard form for easier graphing.
• Rearrange terms to isolate the quadratic expression, keeping in mind that it represents the limits of the function concerning \(y\).

The goal is to find where the inequality "turns" and whether the area of interest lies above or below the curve. Quadratic inequalities require both graphing skills and understanding the behavior of quadratic expressions. Each step helps to visualize the solution set and find the critical solution region.

Parabola Graphing

Graphing a parabola plays a critical role when dealing with quadratic inequalities. Here, the inequality \(y \leq 2 - 6x^2 - x\) corresponds to a downward-opening parabola, discerned from the negative coefficient of the \(-6x^2\) term.
To sketch this parabola:

• Begin by identifying the y-intercept by setting \(x = 0\). For this parabola, the y-intercept is at \((0, 2)\).
• Calculate and plot additional points by choosing various \(x\)-values and solving for \(y\).
• Use these points to draw a symmetrical curve.

Keep in mind that the vertex and the direction of opening (upward or downward) will affect the shading of the solution area. Graphing is essential to visually understand the inequality's solution region.

Solution Set Determination

After graphing the parabola, the next step is to identify and shade the solution set of the inequality. For \(y \leq 2 - 6x^2 - x\), this involves finding the region below or on the parabola since \(\leq\) indicates all y-values equal to or less than those on the boundary.
Shading the parabola:

• Draw the parabola first to mark the boundary.
• Since \(y\) is less than or equal to the expression, shade the area beneath the curve.

This shaded region represents all the solutions that satisfy the inequality. It's a visual representation where any point in this area will make the inequality true when substituted back. Determining solution sets through shading is a powerful technique for grasping algebraic concepts.

Inequality Testing

Testing specific points against the inequality helps determine if they lie within the solution set. For this problem, you need to verify the point \((10, 0)\). Substituting into the inequality \(y \leq 2 - 6x^2 - x\):

• Substitute \(x = 10\) and \(y = 0\).
• Calculate: \(0 \leq 2 - 6(10)^2 - 10\).
• Simplify to see if \(0 \leq -608\), which is clearly false.

Since the result doesn't satisfy the inequality, the point \((10, 0)\) is not within the solution set. Testing is an effective way to confirm whether particular points satisfy the inequality condition or not. By judiciously choosing different points, you can ensure your shaded solution region is correctly represented.