Problem 12 - College Algebra

Question

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ 9 x^{2}+3 x-2 \geq 0 $$

Step-by-Step Solution

Answer

After carrying out the steps, you will have the solution for \( x \) in the form of an interval notation, and you will have a number line graph that represents this solution.

Step 1: Find Critical Points

The critical points, \(x_{1}\) and \(x_{2}\), are found by setting the polynomial equal to zero and solving for \(x\). Therefore, the solutions to \(9x^{2}+3x-2=0\) are calculated using the quadratic formula \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\).

Step 2: Calculate the Test Points

The critical points divide the number line into regions. Select a test point from each region. If \(x1

Step 3: Test the Inequality

Substitute each test point into the inequality \(9x^2+3x-2\geq 0\). If the inequality is valid, include that region in the solution set.

Step 4: Generate the Solution Set in Interval Notation

Express the solution set in interval notation by writing the interval for each included region. Use a square bracket [ or ] to include a point and a parenthesis ( or ) to exclude a point. The solution set includes all points in each interval.

Step 5: Graph the Solution Set

Plot the solution set on a number line. Mark the critical points. Darken the line segment or segments representing the intervals included in the solution set. Leave segments representing excluded intervals unshaded.

Key Concepts

Quadratic Formula

The quadratic formula is a vital tool in algebra for solving quadratic equations, which are polynomial equations of the form \(ax^2 + bx + c = 0\). To find the values of \(x\) that make the equation true, we use the quadratic formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Here's how it works:1. Coefficients Identification: First, identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation. For example, in \(9x^2 + 3x - 2 = 0\), we have \(a = 9\), \(b = 3\), and \(c = -2\).2. Discriminant Calculation: Compute the discriminant \(b^2 - 4ac\), which helps determine the nature of the roots (real or complex).3. Solve for \(x\): Insert \(a\), \(b\), and \(c\) into the formula to find the solutions for \(x\). The signs "\(\pm\)" indicate that you need to consider both the positive and negative square roots when calculating roots.
Understanding how to apply the quadratic formula will give you a systematic method to find solutions for any quadratic equation you encounter.

Test Points

Test points are a strategic approach used to determine which intervals on the number line satisfy a given inequality. After solving the equation using the quadratic formula, you determine critical points, which divide the number line into intervals. Each region between or beyond these critical points provides a potential solution space. 1. Identify Critical Points: These are the solutions found using the quadratic formula. They divide the number line into specific regions. 2. Select Test Points: Choose a test point from each region:

• Choose one point on the left of the smaller root (critical point).
• Choose one point between the roots.
• Choose one point to the right of the larger root.

3. Substitute and Evaluate: Plug each test point back into the original inequality. Check which intervals satisfy the inequality. Pack the understanding that test points reveal if sections of the number line solve the inequalities.
Using test points effectively ensures you include and exclude the correct regions in your solution set.

Solution Set

The solution set is the collection of all values that satisfy the given inequality. When solving a polynomial inequality, like a quadratic one, the solution set consists of the intervals that make the inequality true. - Combine Findings: Once you know which test points satisfy the inequality, you compile these intervals into a coherent solution. - Application to Inequality: Return to your initial inequality. Using test points helps identify which intervals or sections of the number line fulfill the conditions set by the inequality. - Graphical Representation: Visualizing your solution on the number line helps ensure your solution set correctly reflects the inequality’s requirements.
Whether for homework or exams, understanding this concept allows you to convey and understand solutions to polynomial inequalities accurately.

Interval Notation

Interval notation is a concise way to denote a set of numbers belonging to a certain range on the number line, which makes solutions easy to read and understand.- Open and Closed Intervals:

• A closed interval \([a, b]\) includes both endpoints \(a\) and \(b\).
• An open interval \((a, b)\) excludes both endpoints.
• Mixed intervals like \((a, b]\) or \([a, b)\) include one endpoint but not the other.

- Representation of Infinity: Use parentheses when intervals extend to infinity, \((a, \infty)\) or \((-\infty, b]\).
This notation is essential when expressing solution sets. It provides clarity by specifying exactly which parts of the number line are included in the solution.

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Other exercises in this chapter

Problem 11

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$ \frac{4 x^{4}-4 x^{2}+6 x}{x-4} $$

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Problem 11

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Problem 12

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Problem 12

In Exercises \(9-16\) a. List all possible rational zeros. b. Use synthetic division to test the possible rational zeros and find an actual zero. c. Use the quo

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Key Concepts

Quadratic Formula

The quadratic formula is a vital tool in algebra for solving quadratic equations, which are polynomial equations of the form \(ax^2 + bx + c = 0\). To find the values of \(x\) that make the equation true, we use the quadratic formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Here's how it works:1. Coefficients Identification: First, identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation. For example, in \(9x^2 + 3x - 2 = 0\), we have \(a = 9\), \(b = 3\), and \(c = -2\).2. Discriminant Calculation: Compute the discriminant \(b^2 - 4ac\), which helps determine the nature of the roots (real or complex).3. Solve for \(x\): Insert \(a\), \(b\), and \(c\) into the formula to find the solutions for \(x\). The signs "\(\pm\)" indicate that you need to consider both the positive and negative square roots when calculating roots.
Understanding how to apply the quadratic formula will give you a systematic method to find solutions for any quadratic equation you encounter.

Test Points

Test points are a strategic approach used to determine which intervals on the number line satisfy a given inequality. After solving the equation using the quadratic formula, you determine critical points, which divide the number line into intervals. Each region between or beyond these critical points provides a potential solution space. 1. Identify Critical Points: These are the solutions found using the quadratic formula. They divide the number line into specific regions. 2. Select Test Points: Choose a test point from each region:

• Choose one point on the left of the smaller root (critical point).
• Choose one point between the roots.
• Choose one point to the right of the larger root.

3. Substitute and Evaluate: Plug each test point back into the original inequality. Check which intervals satisfy the inequality. Pack the understanding that test points reveal if sections of the number line solve the inequalities.
Using test points effectively ensures you include and exclude the correct regions in your solution set.

Solution Set

The solution set is the collection of all values that satisfy the given inequality. When solving a polynomial inequality, like a quadratic one, the solution set consists of the intervals that make the inequality true. - Combine Findings: Once you know which test points satisfy the inequality, you compile these intervals into a coherent solution. - Application to Inequality: Return to your initial inequality. Using test points helps identify which intervals or sections of the number line fulfill the conditions set by the inequality. - Graphical Representation: Visualizing your solution on the number line helps ensure your solution set correctly reflects the inequality’s requirements.
Whether for homework or exams, understanding this concept allows you to convey and understand solutions to polynomial inequalities accurately.

Interval Notation

Interval notation is a concise way to denote a set of numbers belonging to a certain range on the number line, which makes solutions easy to read and understand.- Open and Closed Intervals:

• A closed interval \([a, b]\) includes both endpoints \(a\) and \(b\).
• An open interval \((a, b)\) excludes both endpoints.
• Mixed intervals like \((a, b]\) or \([a, b)\) include one endpoint but not the other.

- Representation of Infinity: Use parentheses when intervals extend to infinity, \((a, \infty)\) or \((-\infty, b]\).
This notation is essential when expressing solution sets. It provides clarity by specifying exactly which parts of the number line are included in the solution.