Problem 3 - Algebra for College Students

Question

Solve each inequality and graph its solution set on a number line. $$(x+1)(x+4)<0$$

Step-by-Step Solution

Answer

The solution is \((-4, -1)\), and it is graphed as an open interval on the number line.

Step 1: Identify Critical Points

Set each factor of the inequality \((x+1)(x+4) < 0\) to zero to find the critical points. \(x+1 = 0\) gives \(x = -1\) and \(x+4 = 0\) gives \(x = -4\). The critical points divide the number line into intervals.

Step 2: Test Intervals

The critical points \(-4\) and \(-1\) divide the number line into three intervals: \((-\infty, -4)\), \((-4, -1)\), and \((-1, \infty)\). Choose a test point from each interval to determine where the inequality is satisfied. - For \(x = -5\) in \((-\infty, -4)\): \((-5+1)(-5+4) = (-4)(-1) = 4\). This is positive, so the inequality is not satisfied.- For \(x = -2\) in \((-4, -1)\): \((-2+1)(-2+4) = (-1)(2) = -2\). This is negative, so the inequality is satisfied.- For \(x = 0\) in \((-1, \infty)\): \((0+1)(0+4) = (1)(4) = 4\). This is positive, so the inequality is not satisfied.

Step 3: Write the Solution

Since the inequality \((x+1)(x+4) < 0\) is satisfied only within the interval \((-4, -1)\), this is the solution set. The solution is an open interval because neither \(-1\) nor \(-4\) makes the inequality true.

Step 4: Graph the Solution

Draw a number line and place open circles at \(-4\) and \(-1\) since these points are not included in the solution. Shade the region between \(-4\) and \(-1\) to represent the interval \((-4, -1)\) where the inequality holds true.

Key Concepts

Critical Points in Inequalities

When solving inequalities, critical points serve as boundaries that can change the sign of the expression. For an inequality like \[(x+1)(x+4) < 0\]we begin by finding these points. Set the expression equal to zero:

• \(x+1=0\) results in \(x=-1\)
• \(x+4=0\) results in \(x=-4\)

These critical points at \(-4\) and \(-1\) are crucial as they divide the number line into different sections. In each section, the expression can behave differently, sometimes switching signs. Pinpointing these locations helps determine where the expression meets the inequality condition.

Interval Testing

Interval testing is the method used to check which sections defined by critical points satisfy the inequality. After identifying critical points, the number line is split into intervals:

• \((-\infty, -4)\)
• \((-4, -1)\)
• \((-1, \infty)\)

Select a test point from each interval, substitute it into the original inequality:

• For \(x = -5\) in \((-\infty, -4)\), the expression is positive.
• For \(x = -2\) in \((-4, -1)\), it is negative. Thus, the inequality is satisfied here.
• For \(x = 0\) in \((-1, \infty)\), the result turns positive again.

The interval where the expression remains negative is the solution.

Graphing Solution Sets

Visualizing solution sets of inequalities on a graph, specifically a number line, enhances comprehension. Once the valid interval is determined, mark it:

• Use open circles where critical points occur, \(-4\) and \(-1\).
• Draw a line to connect these points.
• Shade the line between \(-4\) and \(-1\) to indicate where the inequality holds true.

This visual gives a clear representation of parts of the number line included in the solution set.

Open Intervals

Solution sets of certain inequalities are termed open intervals. An open interval does not include its endpoints. For this inequality, \((x+1)(x+4) < 0\)tests reveal the inequality holds between critical points \(-4\) and \(-1\), but not at them. Hence, the solution is \((-4, -1)\).

• The endpoints \(-4\) and \(-1\) are not part of the solution, indicated by the lack of equality in the inequality symbol "<".
• Such cases require using open circles in the number line graph.

Open intervals are defined with parentheses \(( )\) to denote exclusion of boundary points.

Number Line Representation

Using a number line is an effective way to represent solutions visually. To construct one for this inequality:

• Draw a horizontal line split into equal sections.
• Mark critical points with open circles to signify they aren’t included in the solution.
• Shade the interval between \(-4\) and \(-1\) to highlight where the inequality \((x+1)(x+4) < 0\)is true.

Number lines provide an intuitive snapshot of valid solutions, ensuring you accurately depict and understand the range satisfying the inequality.