Problem 38 - Beginning and Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write the solution in interval notation. $$(m+4)(m-7)(m+1) \leq 0$$

Step-by-Step Solution

Answer

The solution to the inequality \((m+4)(m-7)(m+1) \leq 0\) is the interval \((-1, 7]\). This can be found by determining the roots (-4, -1, and 7), testing intervals around the roots, and graphing the solution set.

Step 1: Find the Roots

Find the roots by setting each of the linear factors equal to zero: \(m+4 = 0\) \(m = -4\) \(m-7 = 0\) \(m = 7\) \(m+1 = 0\) \(m = -1\) The roots are -4, -1, and 7.

Step 2: Determine the intervals of the domain

Find the intervals on the number line by placing the roots in increasing order: \(-\infty < m \le -4\) \(-4 < m \le -1\) \(-1 < m \le 7\) \(7 < m < \infty\) Since the inequality includes the zero, we use equalities (≤) between the intervals.

Step 3: Test the intervals for the inequality

Test the intervals using sample points from each interval to see if the inequality holds: (1) \(-\infty < m \le -4\) Test a point less than -4, like m = -5: \((-5+4)(-5-7)(-5+1) = (-1)(-12)(-4)\) Positive x Negative x Negative = Positive (Inequality not satisfied) (2) \(-4 < m \le -1\) Test a point between -4 and -1, like m = -2: \((-2+4)(-2-7)(-2+1) = (2)(-9)(-1)\) Positive x Negative x Negative = Positive (Inequality not satisfied) (3) \(-1 < m \le 7\) Test a point between -1 and 7, like m = 0: \(0+4)(0-7)(0+1) = 4(-7)(1)\) Positive x Negative x Positive = Negative (Inequality satisfied) (4) \(7 < m < \infty\) Test a point greater than 7, like m = 8: \((8+4)(8-7)(8+1) = (12)(1)(9)\) Positive x Positive x Positive = Positive (Inequality not satisfied)

Step 4: Write the solution in interval notation and graph the solution set

Only the interval \(-1 < m \le 7\) satisfies the inequality. In interval notation, the solution is: \((-1, 7]\) The graph will be an open circle at -1 and a closed circle at 7, with the line between the two intervals shaded.

Key Concepts

Interval Notation

Interval notation is a helpful way to express the solution of an inequality. It tells us which numbers or intervals are included in the solution. This notation uses parentheses and brackets to show where a sequence starts and ends.

Here's how to understand it:

• Parentheses \((\) or \()\): These indicate that the endpoint is not included in the interval. For example, \((-1, 7)\) means values greater than -1 and less than 7.
• Brackets \([\) or \()]\): These mean the endpoint is included. Like in \([-1, 7]\), both -1 and 7 are part of the solution.

For mathematical inequalities like \((-1,7]\), it is essential to observe the open or closed nature of the endpoints, as they dictate whether the endpoints themselves are part of the solution.

Inequality Graphing

Graphing inequalities can visually show you the solution set on a number line. This method makes it easier to see which parts of the number line satisfy the inequality.

For example, when graphing \(-1 < m \leq 7\), you would:

• Draw a number line and mark the key points identified in the inequalities, such as -1 and 7.
• Place an open circle (indicating exclusion) at -1 since it is not part of the solution \((-1,\).
• Place a closed circle (indicating inclusion) at 7 since the inequality allows for \(\leq 7\), including 7.
• Shade the section of the line between these points to indicate all numbers in this region are solutions to the inequality.

These visual representations make it easy to understand which values satisfy the inequality without switching constantly between algebraic expressions.

Roots of Polynomial Inequalities

To solve polynomial inequalities such as \((m+4)(m-7)(m+1) \leq 0\), finding the roots is the first step. These roots are the values of \(m\) that make each factor in the polynomial equal to zero.

For instance:

• The root of \(m+4=0\) is \(m=-4\).
• The root of \(m-7=0\) is \(m=7\).
• The root of \(m+1=0\) is \(m=-1\).

These roots provide important boundary points to split the number line into intervals, each needing analysis to determine which satisfy the original inequality. By testing values from each interval, you discover which intervals work. This is crucial because polynomials change signs at their roots.

Number Line Analysis

Analyzing a number line helps you determine which intervals satisfy a polynomial inequality. After identifying the roots of the polynomial, you can map them on a number line to divide it into distinct sections or intervals.

Here's the step-by-step:

• Place the roots (-4, -1, and 7) on the number line in increasing order.
• Identify the four resulting intervals:
  • \((-\infty, -4]\)
  • \((-4, -1]\)
  • \((-1, 7]\)
  • \((7, \infty)\)
• Pick a test value from each interval and substitute it into the inequality.
• Determine if the outcome fits the conditions of the inequality.

By examining each interval separately, you decide which sections of the line fulfill the inequality's requirements. Number line analysis is a valuable method for quickly testing and validating the parts of the solution set in polynomial inequalities.