Problem 63 - Algebra for College Students

Question

Solve each inequality. State the solution set using interval notation when possible. \(16-x^{2}>0\)

Step-by-Step Solution

Answer

The solution set is \((-4, 4)\).

Step 1: Rewrite the Inequality

Rewrite the inequality so that all terms are on one side: \[16 - x^2 > 0\]

Step 2: Factor the Expression

Express the left-hand side as a difference of squares: \[16 - x^2 = (4 + x)(4 - x)\]The inequality then becomes: \[(4 + x)(4 - x) > 0\]

Step 3: Determine Critical Points

Identify the values of \(x\) that make the expression equal to zero. Set each factor equal to zero and solve for \(x\):\[4 + x = 0 \ x = -4\]\[4 - x = 0 \ x = 4\]

Step 4: Interval Testing

Test the intervals determined by the critical points to see where the product is positive:1. Interval \((-\infty, -4)\): Pick \(x = -5\), then \ (4 + (-5)) (4 - (-5)) = -1 \times 9 = -9 \text{ (negative)} \[\]2. Interval \((-4, 4)\): Pick \(x = 0\), then \ (4 + 0) (4 - 0) = 4 \times 4 = 16 \text{ (positive)} \[\]3. Interval \((4, \infty)\): Pick \(x = 5\), then \ (4 + 5) (4 - 5) = 9 \times -1 = -9 \text{ (negative)}

Step 5: Conclusion

The inequality \((4 + x)(4 - x) > 0\) is satisfied in the interval \((-4, 4)\). Thus, the solution set in interval notation is: \[(-4, 4)\]

Key Concepts

inequality solving

Solving inequalities is a crucial skill in algebra, much like solving equations. However, solving inequalities involves finding the set of values that make the inequality true. The process generally involves the following steps:

• Rewriting the inequality
• Factoring or simplifying the expression
• Determining critical points
• Testing the intervals between these critical points
• Drawing conclusions based on these tests

interval notation

Once we solve an inequality, we often need to express the solution in a compact form called interval notation. This notation uses parentheses **( )** for open intervals and square brackets **[ ]** for closed intervals. An open interval such as ((-4, 4)), signifies that values approaching -4 and 4 are included but the edge values are not. Conversely, a closed interval like ([-4, 4]), includes the edge values.

difference of squares

The expression (16 - x^2) is a difference of squares, which can be factored. Two squared terms subtracted from each other result in a product of sums and differences: (a^2 - b^2 = (a + b)(a - b)). In this case, it becomes (4^2 - x^2) and is factored as (4 + x)(4 - x). This transformation simplifies solving the inequality.

critical points

Critical points are values that make one of the factors equal to zero. In the inequality (4 + x)(4 - x) > 0), the critical points are found by setting each factor equal to zero: \(4 + x = 0\) and \(4 - x = 0\). Solving these gives critical points: (-4) and (4). These points are crucial because they divide the number line into intervals that we need to test.

interval testing

Interval testing involves picking a test point within each interval created by the critical points and verifying if the inequality holds. For the inequality: (4 + x)(4 - x) > 0), we have three intervals to test:
(-∞, -4), (-4, 4), (4, ∞). We choose a sample number from each interval and plug it back into the inequality:

• (For x = -5, negative result)
• (For x = 0, positive result)
• (For x = 5, negative result)

Since we need a positive result, we conclude the solution interval is ((-4, 4))