Problem 54 - Algebra for College Students

Question

Solve each inequality. $$\frac{2 x-1}{x+2} \geq-1$$

Step-by-Step Solution

Answer

The solution is \((-\infty, -2) \cup \left[-\frac{1}{3}, \infty\right)\).

Step 1: Move All Terms to One Side

To start solving the inequality \(\frac{2x-1}{x+2} \geq -1\), move all terms to one side of the inequality. Begin by adding 1 to both sides to obtain: \[ \frac{2x-1}{x+2} + 1 \geq 0. \]

Step 2: Combine the Fractions

Combine the fractions on the left-hand side. Rewrite 1 as \(\frac{x+2}{x+2}\) to have a common denominator: \[ \frac{2x-1}{x+2} + \frac{x+2}{x+2} = \frac{2x-1 + x+2}{x+2}. \] Simplifying gives: \[ \frac{3x+1}{x+2} \geq 0. \]

Step 3: Identify Critical Points

To determine the critical points, find where the inequality is equal to zero or undefined. Set the numerator equal to zero: \(3x+1 = 0\), solving gives \(x = -\frac{1}{3}\). Set the denominator equal to zero: \(x+2 = 0\), solving gives \(x = -2\). The critical points are \(x = -\frac{1}{3}\) and \(x = -2\).

Step 4: Test Intervals

Determine the sign of the expression \(\frac{3x+1}{x+2}\) over the intervals determined by the critical points: \((-\infty, -2)\), \((-2, -\frac{1}{3})\), and \((-\frac{1}{3}, \infty)\).- For \((-\infty, -2)\), choose \(x = -3\): \(\frac{3(-3)+1}{-3+2} = \frac{-9+1}{-1} = 8\), positive.- For \((-2, -\frac{1}{3})\), choose \(x = -1\): \(\frac{3(-1)+1}{-1+2} = \frac{-3+1}{1} = -2\), negative.- For \((-\frac{1}{3}, \infty)\), choose \(x = 0\): \(\frac{3(0)+1}{0+2} = \frac{1}{2}\), positive.

Step 5: Write the Solution

From the sign analysis, we see that the inequality \(\frac{3x+1}{x+2} \geq 0\) holds for intervals where the expression is positive or zero. Thus, the solution is the union of the intervals where the inequality holds: \((-\infty, -2) \cup \left[-\frac{1}{3}, \infty\right)\). Note that \(x = -2\) is not included since it makes the denominator zero.

Key Concepts

Critical Points

Critical points are key to solving rational inequalities. They are values of the variable where the expression is either zero or undefined. For this exercise, to find these critical points, you follow two main approaches:

• Zero the Numerator: Set the numerator of the rational expression to zero and solve for the variable. This provides points where the expression itself may change its sign. Here, that means solving \(3x + 1 = 0\), which gives \(x = -\frac{1}{3}\).
• Undefined Denominator: Set the denominator to zero to find where the expression becomes undefined. For \(x + 2 = 0\), solving gives \(x = -2\).

Thus, the critical points for our inequality are \(x = -\frac{1}{3}\) and \(x = -2\). These points help divide the number line into intervals for further analysis.

Interval Testing

Interval testing is a method used to determine the sign of an expression over specific intervals. Once you've identified the critical points, you can divide the real number line into intervals. For this problem, the intervals are: \((-\infty, -2)\), \((-2, -\frac{1}{3})\), and \((-\frac{1}{3}, \infty)\).To perform interval testing:

• Select a test point from each interval. It should be a simple number easy to plug into the expression.
• Substitute this number into the rational expression.
• Determine whether the result is positive or negative, which indicates the sign of the expression in that entire interval.

This helps identify where the expression meets the conditions of the inequality, such as being greater than or equal to zero in our case.

Rational Inequalities

Rational inequalities involve expressions made up of polynomials in the form \(\frac{P(x)}{Q(x)}\). Solving these inequalities requires steps that ensure all parts of the inequality are logically addressed:

• Identifying Critical Points: As discussed, find where the rational expression is zero or undefined.
• Rewriting the Inequality: Move all terms to one side, aiming for zero on the other side for easier comparison.
• Simplifying the Expression: Combine like terms or fractions, often by finding a common denominator. This provides a single expression to work with.
• Testing Intervals: Perform sign tests over each divided interval from the number line. This locates where the expression satisfies the inequality condition.
• Writing Solutions: Assemble the intervals where the inequality holds true and ensure you adhere to any exclusions due to undefined points.

These steps guide solving rational inequalities efficiently and reliably, offering a clear visual approach by focusing on critical points and interval testing.