Problem 1 - Algebra A Combined Function

Question

Write the solution set in interval notation. $$ (x+1)(x+5)>0 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, -5) \cup (-1, \infty)\).

Step 1: Identify the critical points

To find the critical points, set each factor of \((x+1)(x+5)\) equal to zero.\[(x+1) = 0 \quad \Rightarrow \quad x = -1\]\[(x+5) = 0 \quad \Rightarrow \quad x = -5\] The critical points are \(x = -1\) and \(x = -5\).

Step 2: Determine intervals based on critical points

The critical points \(-5\) and \(-1\) divide the real number line into three intervals: \((-\infty, -5)\), \((-5, -1)\), and \((-1, \infty)\).

Step 3: Test each interval

Pick a test point from each interval to determine if the product \((x+1)(x+5)\) is positive.For \((-\infty, -5)\), choose \(x = -6\):\[(-6+1)(-6+5) = (-5)(-1) = 5 > 0\] (Positive)For \((-5, -1)\), choose \(x = -3\):\[(-3+1)(-3+5) = (-2)(2) = -4 < 0\] (Negative)For \((-1, \infty)\), choose \(x = 0\):\[(0+1)(0+5) = 1 \times 5 = 5 > 0\] (Positive)

Step 4: Write the solution in interval notation

The solution set where \((x+1)(x+5) > 0\) is for the intervals where the expression is positive.Thus, the solution set in interval notation is \((-\infty, -5) \cup (-1, \infty)\).

Key Concepts

critical points

In mathematics, finding critical points is a crucial step in solving inequalities. Critical points are the values of the variable where the expression could potentially change sign. These are often found by setting each factor of a polynomial expression equal to zero. For the inequality \((x+1)(x+5)>0\), we identify critical points to understand where the expression could be zero or change direction on the real number line.

• The factor \(x+1\) becomes zero when \(x = -1\).
• The factor \(x+5\) becomes zero when \(x = -5\).

These critical points divide the number line into different segments, helping us test each interval's behavior and determine the solution set of the inequality.

inequality solutions

Inequality solutions focus on determining which parts of the real number line satisfy the given inequality. Once critical points are identified, they aid in segmenting the number line into various intervals. This allows for detailed analysis. In our example, \((x+1)(x+5)>0\), the critical points are \(-5\) and \(-1\). These points split the number line into three main intervals:

• \((-\infty, -5)\)
• \((-5, -1)\)
• \((-1, \infty)\)

Each interval has to be tested to see if the inequality holds true within it. You'll find that when the product of terms is positive in an interval, the inequality is satisfied in that region. The intervals can be open-ended or closed depending on whether the inequality is strictly greater than or includes equality (like \(>\) or \(\geq\)). In this exercise, only the intervals where \((x+1)(x+5)\) is greater than zero are part of the solution.

real number line

Visualizing inequalities on the real number line offers a simple yet powerful way to understand solutions. The real number line is a continuous line that includes all possible real numbers, stretching infinitely in both directions. It provides a means to organize and compare values.

When working through inequalities, the real number line is divided using critical points as boundaries. For the problem \((x+1)(x+5)>0\), the critical points are \(-5\) and \(-1\). These create subdivisions that need evaluation. On the number line, you choose various test points in each interval to determine whether the interval satisfies the inequality. It's a clear, strategic method to mark where expressions turn positive, negative, or zero.

interval testing

Interval testing is a strategy used to evaluate where an inequality holds true by choosing random values within intervals defined by critical points. To decide if a solution is part of the solution set, test points are selected from these intervals to see whether they satisfy the inequality.

In the solution \((x+1)(x+5)>0\), testing teaches us which intervals satisfy the inequality:

• For \((-\infty, -5)\), choose \(x = -6\): Plugging into \((x+1)(x+5)\), the expression evaluates to a positive number \(5 > 0\).
• For \((-5, -1)\), choose \(x = -3\): The result is negative \(-4 < 0\).
• For \((-1, \infty)\), choose \(x = 0\): This yields a positive \(5 > 0\).

Intervals yielding a positive result are where the inequality is fulfilled. By continuously repeating the interval testing process for each section, you obtain the solution set for the inequality.