Problem 51 - Algebra for College Students

Question

Solve each inequality. $$\frac{x+2}{x-3}>-2$$

Step-by-Step Solution

Answer

The solution is \((-\infty, \frac{4}{3}) \cup (3, \infty)\).

Step 1: Subtract -2 from both sides

Start by moving the constant to the other side of the inequality to isolate the fraction. Our inequality is \(\frac{x+2}{x-3} > -2\).Subtract \(-2\) from both sides to rewrite the inequality:\[\frac{x+2}{x-3} + 2 > 0\]

Step 2: Combine the terms into a single fraction

Combine the terms on the left side into a single fraction. Start by writing 2 as a fraction over \(x-3\), yielding:\[ \frac{x+2 + 2(x-3)}{x-3} > 0 \]Simplify the numerator:\[ \frac{x+2 + 2x - 6}{x-3} > 0 \]Combine like terms:\[ \frac{3x - 4}{x-3} > 0 \]

Step 3: Determine critical points

To solve \(\frac{3x - 4}{x-3} > 0\), find when the expression equals zero or is undefined. Set the numerator and denominator each equal to zero to find the critical points.1. \(3x - 4 = 0\) - Solve for \(x\): \[3x = 4 \rightarrow x = \frac{4}{3}\]2. \(x - 3 = 0\) - Solve for \(x\): \[x = 3\]The critical points are \(x = \frac{4}{3}\) and \(x = 3\).

Step 4: Construct a sign chart

Use the critical points to divide the number line into intervals and test the inequality in each interval. The intervals are: 1. \(x < \frac{4}{3}\)2. \(\frac{4}{3} < x < 3\)3. \(x > 3\)Choose test points within each region:- For \(x < \frac{4}{3}\), use \(x = 0\): \(\frac{3(0) - 4}{0 - 3} = \frac{-4}{-3} = \frac{4}{3} > 0\) Valid- For \(\frac{4}{3} < x < 3\), use \(x = 2\): \(\frac{3(2) - 4}{2 - 3} = \frac{2}{-1} = -2 < 0\) Invalid- For \(x > 3\), use \(x = 4\): \( \frac{3(4) - 4}{4 - 3} = \frac{8}{1} = 8 > 0\) Valid

Step 5: Write the solution

From the sign chart, the inequality is true for \(x < \frac{4}{3}\) and \(x > 3\). The solution set in interval notation is:\((-\infty, \frac{4}{3}) \cup (3, \infty)\).

Key Concepts

Critical Points

In mathematical inequalities, critical points are values that influence the sign of an expression. They occur where the expression equals zero or becomes undefined. Identifying these points is essential since they help determine where an inequality switches from positive to negative, or vice versa.
To find critical points:

• Set the numerator of your fraction equal to zero. Solve for the variable, and the resulting value is a critical point.
• Set the denominator equal to zero because division by zero makes the fraction undefined. Solve the equation, and this provides another critical point.

For the inequality \(\frac{3x - 4}{x-3} > 0\), the critical points are found by setting \(3x - 4 = 0\) and \(x - 3 = 0\). Solving these gives us the critical points \(x = \frac{4}{3}\) and \(x = 3\). These values help divide the number line into regions for analysis.

Interval Notation

Interval notation is a concise way of representing solutions to inequalities, using intervals to display where the inequality holds true on the number line. This method visually conveys whether the endpoints themselves are included or excluded.
In interval notation:

• Parentheses \(( )\) indicate that an endpoint is not included in the interval (open interval).
• Square brackets \([ ]\) indicate that an endpoint is included (closed interval).

For the solution where the expression \(\frac{3x - 4}{x-3} > 0\) holds true, the intervals are \((-\infty, \frac{4}{3})\) and \((3, \infty)\). These exclude the critical points \(x = \frac{4}{3}\) and \(x = 3\) where the inequality does not hold, thus using parentheses.

Sign Chart

A sign chart is a powerful tool used to determine where an inequality holds true by testing intervals created by critical points on the number line. It allows us to assess the sign (positive or negative) of the expression in each interval.
Constructing a sign chart involves:

• Identifying critical points and plotting them on a number line to split it into intervals.
• Selecting test points within each interval to substitute into the inequality.
• Determining the sign of the expression (e.g., positive or negative) based on the result.

For the inequality \(\frac{3x - 4}{x-3} > 0\), critical points \(x = \frac{4}{3}\) and \(x = 3\) divide the number line into three segments. By testing a point in each interval:

• In \(x < \frac{4}{3}\), the test point \(x = 0\) results in a positive sign.
• In \(\frac{4}{3} < x < 3\), the test point \(x = 2\) yields a negative sign.
• In \(x > 3\), the test point \(x = 4\) results in a positive sign.

The valid regions where the inequality \(\frac{3x - 4}{x-3} > 0\) holds true are from \(-\infty\) to \(\frac{4}{3}\) and from \(3\) to \(\infty\). This information is crucial for writing the solution in interval notation.