Problem 40 - College Algebra with Modeling and Visualization

Question

Solve the inequality. $$ (x-3.1)(x+2.7)>0 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -2.7) \cup (3.1, +\infty)\).

Step 1: Identify the Critical Points

The given inequality is \[(x - 3.1)(x + 2.7) > 0.\]To find the critical points where the expression will be equal to zero, set each factor equal to zero. Solve \(x - 3.1 = 0\) and \(x + 2.7 = 0\), giving critical points \(x = 3.1\) and \(x = -2.7\). These are the points where the sign of the expression can potentially change.

Step 2: Analyze the Intervals

The critical points divide the number line into three intervals: \((-\infty, -2.7)\), \((-2.7, 3.1)\), and \((3.1, +\infty)\).For each interval, pick a test point and determine if the product \((x-3.1)(x+2.7)\) is positive or negative.

Step 3: Test Interval \\( (-\infty, -2.7) \\)

Choose a test point such as \(x = -3\). Substituting into \((x - 3.1)(x + 2.7)\) gives \((-3 - 3.1)(-3 + 2.7) = (-6.1)(-0.3) > 0\). Thus, the interval \((-\infty, -2.7)\) satisfies the inequality.

Step 4: Test Interval \\( (-2.7, 3.1) \\)

Choose a test point such as \(x = 0\). Substituting into \((x - 3.1)(x + 2.7)\) gives \((0 - 3.1)(0 + 2.7) = (-3.1)(2.7) < 0\). Hence, \((-2.7, 3.1)\) does not satisfy the inequality.

Step 5: Test Interval \\( (3.1, +\infty) \\)

Choose a test point such as \(x = 4\). Substituting into \((x - 3.1)(x + 2.7)\) gives \((4 - 3.1)(4 + 2.7) = (0.9)(6.7) > 0\). Thus, the interval \((3.1, +\infty)\) satisfies the inequality.

Step 6: Combine the Results

The solution includes intervals where the inequality holds true, which are \((-\infty, -2.7)\cup (3.1, +\infty)\). The critical points \(x = -2.7\) and \(x = 3.1\) are not included since the inequality is strict (i.e., \(> 0\)).

Key Concepts

Critical Points

Critical points are essential in understanding inequalities because they identify where the expression changes its sign—from positive to negative or vice versa. To find critical points in the inequality \[(x - 3.1)(x + 2.7) > 0,\]we set each factor equal to zero:

• First factor: \(x - 3.1 = 0\)
• Second factor: \(x + 2.7 = 0\)

Solving these equations gives us the points \(x = 3.1\) and \(x = -2.7\).
These are the boundaries where the expressions can change sign. Understanding critical points helps us segment the number line effectively for further analysis.
Choosing test points from each segment allows us to determine the truth of the inequality within each interval.

Interval Testing

Interval testing helps in determining whether an algebraic expression is positive or negative in certain sections of the number line. For the inequality \[(x - 3.1)(x + 2.7) > 0,\]we divide the number line using our critical points:

• \((-\infty, -2.7)\)
• \((-2.7, 3.1)\)
• \((3.1, \infty)\)

For each interval, choose a sample point, substitute it into \((x - 3.1)(x + 2.7),\) and check the sign:

• In \((-\infty, -2.7),\)picking \(x = -3\) results in a positive value;
• In \((-2.7, 3.1),\)picking \(x = 0\) results in a negative value;
• In \((3.1, \infty),\)picking \(x = 4\) results in a positive value.

Analyzing these intervals helps build a picture of where the inequality holds true across the number line.

Algebraic Expressions

Algebraic expressions are the foundation for constructing inequalities like \((x - 3.1)(x + 2.7) > 0.\)They consist of operations involving variables and constants. Manipulating these expressions correctly is the key to solving inequalities.
In this equation, multiplication of the factors \((x - 3.1)\) and \((x + 2.7)\)forms the expression on the left side of the inequality. When we solve for these expressions, we are interested in where their combined product becomes greater than zero.
Recognizing that expressions can be factored into simpler binomials allows us to easily find critical points and test intervals.

Solution Sets

Solution sets are the final answer when solving inequalities. They tell us precisely which values of \(x\) satisfy the inequality. Once interval testing is complete, we combine the results to determine the full solution set.
For \((x - 3.1)(x + 2.7) > 0,\)we found that the intervals \((-\infty, -2.7)\) and \((3.1, \infty)\)make the product positive. Critical points \(-2.7\) and \(3.1\)are not included, since our inequality is strict, denoted by ">" rather than "≥".
Thus, the solution set is expressed as \((-\infty, -2.7) \cup (3.1, \infty).\)This notation includes all numbers in those intervals, but not the individual critical points.