Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-40: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ \frac{x-2}{3 x+5} \leq 4 $$

Step-by-Step Solution

Answer

The solution is \( x \in (-\infty, -2] \).

Step 1: Understand the Inequality

The given inequality is \( \frac{x-2}{3x+5} \leq 4 \). We aim to solve this inequality for \( x \) and express the solution as an interval, if possible.

Step 2: Move All Terms to One Side

Subtract 4 from both sides to isolate the rational expression on one side of the inequality: \( \frac{x-2}{3x+5} - 4 \leq 0 \).

Step 3: Simplify the Expression

Rewrite \( 4 \) as a fraction \( \frac{4(3x+5)}{3x+5} \) so we can combine the fractions: \( \frac{x-2}{3x+5} - \frac{4(3x+5)}{3x+5} \leq 0 \). Combine the fractions to get: \( \frac{x-2 - 4(3x+5)}{3x+5} \leq 0 \).

Step 4: Simplify the Numerator

Distribute the \( -4 \) in the numerator: \( x-2 - 12x - 20 = -11x - 22 \). The inequality is now \( \frac{-11x-22}{3x+5} \leq 0 \).

Step 5: Identify Critical Points

Set the numerator \( -11x-22 = 0 \) to find points where the expression changes sign. Solving gives \( x = -2 \). Set the denominator \( 3x+5 = 0 \) to find points of undefined values, giving \( x = -\frac{5}{3} \).

Step 6: Test Intervals Around Critical Points

Test intervals: - For \( x < -2 \), choose \( x = -3 \): \( \frac{-11(-3)-22}{3(-3)+5} = \frac{33-22}{-9+5} = \frac{11}{-4} < 0 \), so this interval satisfies the inequality.- For \( -2 < x < -\frac{5}{3} \), choose \( x = -1.8 \): Check that this gives a positive value.- For \( x > -\frac{5}{3} \), choose \( x = 0 \): Check that this gives a positive value.

Step 7: Express the Solution

The solution is \( x \in (-\infty, -2] \). The critical point \( x = -2 \) is included because the inequality is non-strict (i.e., \( \leq \)).

Key Concepts

Solving Inequalities

When solving inequalities, the goal is to determine the range of values for a variable that satisfy the inequality. In our exercise, we have the inequality \( \frac{x-2}{3x+5} \leq 4 \). To solve it, we need to perform several steps:

• Isolate the rational expression: Begin by moving all terms to one side of the inequality. This often involves basic arithmetic operations like addition or subtraction of the same value on both sides.
• Simplify the expression: Rewrite constants such as integers into a common denominator to allow combining ratios. In this case, transform 4 into \( \frac{4(3x+5)}{3x+5} \).
• Simplify the function: Combine fractions and simplify, paying special attention to the signs and maintaining inequality directions.

Once simplified, solving inequalities often involves finding the critical points and evaluating the intervals they create. This process reveals the solution set easiest visualized in graphical or tabular form. Understanding these solutions relies on analyzing sign changes across critical points, aiding in determining which intervals satisfy the original inequality.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator contain polynomials. In the given exercise, our rational expression is \( \frac{x-2}{3x+5} \). Solving inequalities with rational expressions requires several considerations:

• Identifying Critical Points: Determine where the expression equals zero by setting the numerator equal to zero. Additionally, identify points of discontinuity, typically where the denominator is zero.
• Simplifying Rational Expressions: Often, you'll need to perform operations like cancelling terms when possible. However, simplifying expressions must be done cautiously to avoid altering the inequality’s nature.

The process usually hinges on finding solutions within these expressions' domains and addresses how different values of \(x\) may affect the expression's positivity or negativity. The critical points mark boundaries between regions where the sign of the expression may change, helping in discovering solution intervals.

Interval Notation

Interval notation is a shorthand way to express which portions of the number line are included in the solution. Once the solution for an inequality is determined, it can often be expressed using this notation.

• Understanding Intervals: Parentheses \(( )\) indicate that an endpoint is not included in the interval, known as an open interval. Conversely, brackets \([ ]\) indicate that an endpoint is included, known as a closed interval.
• Writing Solutions in Interval Notation: After determining the intervals from the inequality's critical points, decide which inequalities or ranges do match the conditions given (inclusion or exclusion of endpoints).

For our exercise, after evaluating the critical points, the solution is expressed as \( x \in (-\infty, -2] \). The interval uses \(-\infty\), symbolizing that there's no lowest boundary, and a bracket with -2 showing that this endpoint is included in the solution because the inequality involves "less than or equal to." Understanding this succinct format is crucial for communicating solution sets effectively.