Problem 22 - Intermediate Algebra

Question

Solve each inequality. $$ x(7 x+40)<12 $$

Step-by-Step Solution

Answer

The solution is \( x \in (-6, \frac{2}{7}) \).

Step 1: Expand the Inequality

First, we need to expand the left side of the inequality. Multiply the expression inside the parenthesis by the external factor: \[ x(7x + 40) < 12 \]This yields: \[ 7x^2 + 40x < 12 \].

Step 2: Move All Terms to One Side

To simplify solving for \( x \), place all terms on one side of the inequality to form a quadratic inequality: \[ 7x^2 + 40x - 12 < 0 \].

Step 3: Factor the Quadratic Expression

We need to factor the quadratic equation \( 7x^2 + 40x - 12 = 0 \). To do this, find two numbers that multiply to \( -84 \) (the product of \( 7 \) and \( -12 \)) and add to \( 40 \). Those numbers are \( 42 \) and \( -2 \), so we'll rewrite the middle term as follows:\[ 7x^2 + 42x - 2x - 12 = 0 \].

Step 4: Apply the Factor by Grouping Method

Group the terms and factor by grouping:1. Group the terms: \( (7x^2 + 42x) + (-2x - 12) \)2. Factor each group: \( 7x(x + 6) - 2(x + 6) \)3. Combine to factor completely: \( (x + 6)(7x - 2) = 0 \).

Step 5: Find the Roots

Set each factor equal to zero to find the roots:1. \( x + 6 = 0 \Rightarrow x = -6 \)2. \( 7x - 2 = 0 \Rightarrow 7x = 2 \Rightarrow x = \frac{2}{7} \).

Step 6: Determine the Interval of Solutions

The roots \( x = -6 \) and \( x = \frac{2}{7} \) divide the x-axis into intervals. Test points in each interval \((-\infty, -6)\), \((-6, \frac{2}{7})\), and \((\frac{2}{7}, \infty)\) to determine where the inequality is less than zero:\- Choose \( x = -7 \) for \( (-\infty, -6) \): \( 7(-7)^2 + 40(-7) - 12 > 0 \) (False)- Choose \( x = 0 \) for \( (-6, \frac{2}{7}) \): \( 7(0)^2 + 40(0) - 12 < 0 \) (True)- Choose \( x = 1 \) for \((\frac{2}{7}, \infty)\): \( 7(1)^2 + 40(1) - 12 > 0 \) (False)

Step 7: Write the Solution Set

Since the inequality is true in the interval \((-6, \frac{2}{7})\), the solution to the inequality is:\[ x \in (-6, \frac{2}{7}) \].

Key Concepts

Factoring Quadratics

Factoring quadratics is a straightforward and essential step in solving quadratic inequalities. In our case, we started with the quadratic inequality \(7x^2 + 40x - 12 < 0\). The purpose of factoring is to express the quadratic in a simpler form, typically as a product of binomials. This helps identify critical points, known as the roots or zeros of the quadratic.

Here's how you can approach it:

• First, identify two numbers that multiply to the product of the coefficient of \(x^2\) (which is 7) and the constant term (which is -12). This means looking for numbers that multiply to -84.
• These numbers also need to add up to the coefficient of the linear term (which is 40). For our example, those numbers are 42 and -2.
• Rewrite the middle term (40x) using these numbers: \(7x^2 + 42x - 2x - 12 = 0\).
• Then, apply grouping: group the first two and the last two terms separately.
• Factor each group: \(7x(x + 6) - 2(x + 6)\).
• Finally, factor out the common factor \((x + 6)\) to obtain \((x + 6)(7x - 2)\).

Factoring not only simplifies the expression but also reveals the roots where the quadratic equation changes signs.

Interval Testing

Once you've factored the quadratic, you need to identify the solution intervals by testing values around the roots. This process ensures that you determine where the inequality holds true.

Here's how interval testing works:

• The roots \(x = -6\) and \(x = \frac{2}{7}\) divide the x-axis into different intervals: \((-\infty, -6)\), \((-6, \frac{2}{7})\), and \((\frac{2}{7}, \infty)\).
• Select a test point from each interval and substitute it back into the original inequality \(7x^2 + 40x - 12\).
• For \((-\infty, -6)\), choose \(x = -7\). The result is greater than zero, meaning the inequality isn't satisfied here.
• For \((-6, \frac{2}{7})\), use \(x = 0\), which results in a negative solution, indicating this interval satisfies the inequality.
• Lastly, in \((\frac{2}{7}, \infty)\), try \(x = 1\). The outcome is positive, not satisfying the inequality.

This method is critical as it confirms the correct interval and confirms that you understand where the inequality holds.

Inequality Solutions

Finally, inequality solutions help you interpret the mathematical results into meaningful answers. After factoring and interval testing, you can determine the suitable values of \(x\) that satisfy the inequality.

The main steps are:

• Recognize that the roots \(x = -6\) and \(x = \frac{2}{7}\) do not themselves satisfy the inequality \(7x^2 + 40x - 12 < 0\).
• From interval testing, discover that the interval \((-6, \frac{2}{7})\) satisfies the inequality.
• Write the solution set using interval notation: \(x \in (-6, \frac{2}{7})\).

Inequalities are not just about solving equations. They tell us about ranges of values that fulfill specific conditions, offering a deeper insight into the mathematical relationship expressed.