Problem 10 - Algebra for College Students

Question

Solve each inequality and graph its solution set on a number line. $$(x+2)(x+1)(x-2)>0$$

Step-by-Step Solution

Answer

The solution set is \((-\infty, -2)\cup(2, \infty)\).

Step 1: Identify Critical Points

First, identify the critical points where each factor of the inequality \((x+2)(x+1)(x-2)>0\) equals zero. These are the points where the product could change sign. Solve for each factor: \(x+2=0\) gives \(x=-2\), \(x+1=0\) gives \(x=-1\), and \(x-2=0\) gives \(x=2\). Therefore, the critical points are \(-2, -1,\) and \(2\).

Step 2: Test Intervals Defined by Critical Points

Use the critical points to divide the number line into intervals: \((-\infty, -2),\)\((-2, -1),\) \((-1, 2),\) and \((2, \infty)\). Test each interval by choosing a test point and substituting it into the inequality: - Test point \(x = -3\) for \((-\infty, -2)\) gives positive product: \((-3+2)(-3+1)(-3-2) = (-1)(-2)(-5)>0\).- Test point \(x = -1.5\) for \((-2, -1)\) gives negative product: \((-1.5+2)(-1.5+1)(-1.5-2) = (0.5)(-0.5)(-3.5)<0\).- Test point \(x = 0\) for \((-1, 2)\) gives negative product: \((0+2)(0+1)(0-2) = (2)(1)(-2)<0\).- Test point \(x = 3\) for \((2, \infty)\) gives positive product: \((3+2)(3+1)(3-2) = (5)(4)(1)>0\).

Step 3: Determine Solution Set

The solution to the inequality is where the product is positive. From the tests conducted: - Interval \((-\infty, -2)\) satisfies the inequality.- Interval \((2, \infty)\) satisfies the inequality as well.Therefore, the solution set is \((-\infty, -2)\cup(2, \infty)\).

Step 4: Graph Solution on Number Line

Draw a number line and mark the critical points \(-2\) and \(2\). Use open circles at these critical points since they are not included in the solution (the inequality is '>' and not '>='). Shade the regions \((-\infty, -2)\) and \((2, \infty)\) to represent the solution set where the inequality holds. It appears as two separate shaded regions on the number line.

Key Concepts

Critical Points

In solving inequalities, critical points are essential markers that indicate where an expression might change its sign. For the expression \((x+2)(x+1)(x-2)>0\), the critical points occur where each factor equals zero. This is because at zero, the product of these factors is neither positive nor negative—it is precisely where it might switch its sign.

To find these points, we solve each equation separately:

• For \(x+2=0\), solve to get \(x=-2\);
• For \(x+1=0\), solve to get \(x=-1\);
• For \(x-2=0\), solve to get \(x=2\).

These solutions \(-2, -1,\) and \(2\) are our critical points. They help divide the number line into sections where we can test the inequality's truth.

Test Intervals

Once critical points are identified, we use them to split the number line into distinct intervals. For the inequality \((x+2)(x+1)(x-2)>0\), these intervals become:

• From \(-\infty\) to \(-2\),
• from \(-2\) to \(-1\),
• from \(-1\) to \(2\),
• and from \(2\) to \(+\infty\).

To determine the sign of the expression in each interval, choose any test point within each section and substitute into the inequality.

For example:

• Test point \(x = -3\) gives positive value in the \((-\infty, -2)\) interval.
• Test point \(x = -1.5\) gives negative value in the \((-2, -1)\) interval.
• Test point \(x = 0\) gives negative value in the \((-1, 2)\) interval.
• Test point \(x = 3\) gives positive value in the \((2, \infty)\) interval.

This process helps determine where the inequality holds or fails for each interval.

Solution Set

After testing intervals, the solution set comprises the intervals where the inequality is true (i.e., it results in a positive product). From the tested sections in the example, the inequality holds for:

• The interval \((-\infty, -2)\), where we found the product is positive,
• and the interval \((2, \infty)\), where the product is also positive.

Consequently, the solution set is written as the union of these intervals: \((-\infty, -2)\cup(2, \infty)\). This indicates where on the number line the original inequality \((x+2)(x+1)(x-2)>0\) is satisfied.

Number Line

To visualize the solution set, a number line can be very effective. Firstly, plot the critical points: \(-2\) and \(2\). Because the inequality is strictly greater than (\(>0\) as opposed to \(\geq0\)), these critical points themselves are not included in the solution, necessitating open circles over them.

Next, shade the number line at the intervals \((-\infty, -2)\) and \((2, \infty)\) to depict where the inequality holds true. These shaded sections indicate regions where values of \(x\) make the inequality correct. This graphical representation helps correlate the algebraic solution to a visual format, making it easier to comprehend.