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Magazine Chapter 3: Problem 83
Solve each inequality by using the method of your choice. State the solution set in interval notation and graph it. $$t^{2} \leq 16$$
Short Answer
Expert verified
[-4, 4]
Step by step solution
01
- Move All Terms to One Side
Rewrite the inequality to move all terms to one side: \[ t^2 - 16 \ \text{So it becomes:} \ t^2 - 16 \.leq 0 \]
02
- Factor the Inequality
Factor the quadratic expression on the left-hand side: \[ t^2 - 16 = (t - 4)(t + 4) \ \text{So the inequality now is:} \ (t - 4)(t + 4) \leq 0 \]
03
- Find the Critical Points
Determine the values of t that make each factor equal to zero: \[ t - 4 = 0 \ t = 4 \] \[ t + 4 = 0 \ t = -4 \]
04
- Test Intervals
Test values in each interval created by the critical points (-鈭�, -4), (-4, 4), and (4, 鈭�) to determine where the inequality holds: \[ \text{For example, choose } t = -5, 0, \text{and } 5: \] \[ t = -5 \text{ (choose a point in (鈭掆垶,-4)}): \] \[ (-5 - 4)(-5 + 4) = (-9)(-1) > 0 \] \[ t = 0 \text{ (choose a point in (鈭�4,4)}): \] \[ (0 - 4)(0 + 4) = (-4)(4) < 0 \] \[ t = 5 \text{ (choose a point in (4,鈭�)}): \] \[ (5 - 4)(5 + 4) = (1)(9) > 0 \]
05
- Write the Solution Set in Interval Notation
Combine the intervals where the inequality holds true and include the points where the expression equals zero: \[ t^2 \leq 16 \text{ holds for -4 鈮� t 鈮� 4. In interval notation, this is:} \] \[ \text{[-4, 4]} \]
06
- Graph the Solution
Graph the solution on a number line by shading the region between -4 and 4, including the points -4 and 4 as closed circles to indicate they are included in the solution set.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Quadratic Expressions
To tackle quadratic inequalities, you often start by factoring quadratic expressions. This means rewriting the quadratic expression as the product of two binomial expressions. For example, consider the quadratic expression from our problem: \ t^2 - 16. \ We can factor this expression into: \ (t - 4)(t + 4). \ Factoring helps in simplifying the inequality and finding the critical points. A clue is to look for differences of squares, such as \(a^2 - b^2\), that factor into \((a - b)(a + b)\). Remember to practice factoring thoroughly as it lays the foundation for solving the inequality.
Interval Notation
Once you solve the inequality, you'll need to express the solution set in interval notation. This is a succinct way of describing which parts of the number line satisfy the inequality. For instance, in our problem, the solution set was between -4 and 4, inclusive. \ Thus, in interval notation, it is written as \([-4, 4]\).
The square brackets [ and ] denote that the endpoints -4 and 4 are included (they satisfy the inequality). By contrast, parentheses ( and ) are used when endpoints are not included. Mastering interval notation is essential for clearly communicating the solution.
The square brackets [ and ] denote that the endpoints -4 and 4 are included (they satisfy the inequality). By contrast, parentheses ( and ) are used when endpoints are not included. Mastering interval notation is essential for clearly communicating the solution.
Critical Points
Critical points are the values that make the polynomial in the inequality equal to zero. To find them, you set each factor from the factored quadratic expression to zero and solve for the variable. In our scenario with \( (t - 4)(t + 4) \), solving these gives: \ t - 4 = 0 \ t = 4 \ \ and \ \ t + 4 = 0 \ t = -4.
These points divide the number line into intervals. The critical points are crucial as they help us determine where the inequality is satisfied. Knowing how to find and use critical points is a key skill in solving any quadratic inequality.
These points divide the number line into intervals. The critical points are crucial as they help us determine where the inequality is satisfied. Knowing how to find and use critical points is a key skill in solving any quadratic inequality.
Testing Intervals
After finding the critical points, you need to test the intervals they create to check where the quadratic inequality holds. For our inequality, the critical points \(-4 \text{ and } 4\) created these intervals: (-鈭�, -4), (-4, 4), and (4, 鈭�).
Next, choose a test point in each interval and substitute it into the inequality: \ \ -5 in (-鈭�, -4): \ (-5 - 4)(-5 + 4) = (鈭�9)(鈭�1) > 0 (so, doesn't hold). \ \ 0 in (-4, 4): \ (0 - 4)(0 + 4) = (鈭�4)(4) < 0 (so, holds). \ \ 5 in (4, 鈭�): \ (5 - 4)(5 + 4) = (1)(9) > 0 (so, doesn't hold). \ \ Interval testing ensures that you accurately determine the solution set for the inequality.
Next, choose a test point in each interval and substitute it into the inequality: \ \ -5 in (-鈭�, -4): \ (-5 - 4)(-5 + 4) = (鈭�9)(鈭�1) > 0 (so, doesn't hold). \ \ 0 in (-4, 4): \ (0 - 4)(0 + 4) = (鈭�4)(4) < 0 (so, holds). \ \ 5 in (4, 鈭�): \ (5 - 4)(5 + 4) = (1)(9) > 0 (so, doesn't hold). \ \ Interval testing ensures that you accurately determine the solution set for the inequality.
