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Chapter 1: Problem 53

Nonlinear Inequalities Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$(x-4)(x+2)^{2} < 0$$

Short Answer

Expert verified
The solution is \((-\infty, -2) \cup (-2, 4)\) in interval notation.

Step by step solution

01

Determine the Critical Points

First, find the points where the expression \((x-4)(x+2)^{2} = 0\).These are the values of \(x\) which make the expression equal to zero. Solving \((x-4) = 0\) gives \(x = 4\), and solving \((x+2)^{2} = 0\) gives \(x = -2\), so our critical points are \(x = 4\) and \(x = -2\).
02

Test the Intervals

The critical points divide the real number line into intervals. These intervals are \((-\infty, -2)\), \((-2, 4)\), and \((4, \infty)\). Choose a test point from each interval to determine whether the expression is positive or negative in that interval.- Test \(x = -3\) in \((-\infty, -2)\):\((x-4)(x+2)^{2} = (-3-4)((-3)+2)^{2} = (-7)(1)^{2} = -7\)- Test \(x = 0\) in \((-2, 4)\):\((x-4)(x+2)^{2} = (0-4)((0)+2)^{2} = (-4)(4) = -16\)- Test \(x = 5\) in \((4, \infty)\):\((x-4)(x+2)^{2} = (5-4)((5)+2)^{2} = (1)(7)^{2} = 49\)
03

Determine the Sign of Each Interval

From our test points, we conclude:- The interval \((-\infty, -2)\) is negative- The interval \((-2, 4)\) is negative- The interval \((4, \infty)\) is positive
04

Find the Solution Set

We are looking for where the expression \((x-4)(x+2)^{2} < 0\). This means we need to consider the intervals where the expression is negative.From Step 3, the intervals \((-\infty, -2)\) and \((-2, 4)\) satisfy this condition.Therefore, the solution set in interval notation is:\((-\infty, -2) \cup (-2, 4)\).
05

Graph the Solution Set

The solution includes all real numbers in the intervals \((-\infty, -2)\) and \((-2, 4)\). On a number line, this is represented as a ray going to the left from \(-2\) and a segment between \(-2\) and \(4\). Use open circles at \(-2\) and \(4\) to indicate these points are not included in the solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Critical Points
Critical points are where the expression equals zero. In our inequality \((x-4)(x+2)^2 < 0\), we set the expression equal to zero to find these points. Solving \((x-4) = 0\) gives us \(x=4\), and solving \((x+2)^2 = 0\) gives us \(x=-2\). These values of \(x\) are known as critical points.

Here's why they matter:
  • Critical points divide the number line into intervals where the expression might change from positive to negative or vice versa.
  • These points help us test the intervals to find where the inequality holds true.
Understanding critical points is key in solving inequalities because they indicate possible changes in sign, telling us where to look for solutions.
Interval Notation
Interval notation provides a compact way to express ranges of numbers. In the solution provided, interval notation is used to show where the inequality \((x-4)(x+2)^2 < 0\) is true.

For example:
  • The interval \((-\infty, -2)\) suggests all numbers less than \(-2\).
  • The interval \((-2, 4)\) indicates numbers between \(-2\) and \(4\) but not including either end.
This notation helps in clearly defining the solution in a concise manner. It tells us precisely where, on the number line, the inequalities hold true. Note that parentheses are used in these intervals because the endpoints \(-2\) and \(4\) are not included in the solution.
Solution Set
A solution set is a collection of all solutions that satisfy the inequality. In this context, it tells us the values of \(x\) where the inequality \((x-4)(x+2)^2 < 0\) is valid.

From our solution:
  • The solution set is \((-\infty, -2) \cup (-2, 4)\).
  • The union symbol \(\cup\) signifies that solutions are found in either of the intervals.
The solution set forms a key part of solving inequalities, as it narrows down the possible values of \(x\) to those that make the inequality true.
Real Number Line
The real number line is a visual representation of numbers, extending infinitely in both directions. This line helps in representing and understanding intervals and critical points.

In the context of our inequality, the real number line is divided into segments based on the critical points we found, \(-2\) and \(4\).
  • These points split the number line into regions: \((-\infty, -2)\), \((-2, 4)\), and \((4, \infty)\).
  • We test each of these regions to see if the inequality holds.
  • The number line helps visualize which segments are part of the solution set.
By plotting the solution on a number line, we visually see the real numbers that satisfy the inequality, enhancing understanding of the solution.

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Most popular questions from this chapter

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