Problem 40 - Algebra A Combined Function

Question

Solve each inequality. Write the solution set in interval notation. $$ \frac{4}{y+2}<-2 $$

Step-by-Step Solution

Answer

Interval notation: \((-\infty, -4)\).

Step 1: Isolate the variable

Start by isolating the fraction that contains the variable \( y \). Subtract \( -2 \) from both sides of the inequality: \( \frac{4}{y+2} + 2 < 0 \).

Step 2: Clear the fraction

Multiply every term by \( y+2 \) to eliminate the fraction. Be cautious of the sign change depending on the value of \( y+2 \): \( 4 < -2(y+2) \).

Step 3: Simplify the inequality

Distribute \( -2 \) on the right side: \( 4 < -2y - 4 \).

Step 4: Combine like terms

Add \( 4 \) to both sides to start combining terms: \( 4 + 4 < -2y \). This simplifies to \( 8 < -2y \).

Step 5: Solve for \( y \)

Divide both sides by \(-2\). Remember that dividing by a negative number reverses the inequality sign: \( y < -4 \).

Step 6: Write the solution in interval notation

The solution \( y < -4 \) is expressed in interval notation as \((-\infty, -4)\).

Key Concepts

Interval Notation

Interval notation is a system used to express the set of solutions for an inequality clearly and concisely. Instead of listing all the numbers that satisfy an inequality, we use intervals to represent this range. Here's how interval notation works:

• Round brackets \((\) or \()\) are used to denote that the endpoint numbers are not included in the interval, which corresponds to "less than" (<) or "greater than" (>) in inequalities.
• Square brackets \([\) or \()]\) indicate that the endpoints are included, which corresponds to "less than or equal to" (≤) or "greater than or equal to" (≥).

For example, the solution \( y < -4 \) is represented in interval notation as \((-fty, -4)\), meaning all numbers less than -4 are solutions. The infinite symbol \(fty\) always uses round brackets because infinity is not a fixed number and cannot include endpoints.

Fractional Inequalities

Fractional inequalities involve expressions where the variable appears in the denominator of a fraction. Solving them requires careful steps to avoid common mistakes. Here's what to consider:

• The main goal is to eliminate the fraction to simplify the inequality. This often involves multiplying both sides by the denominator to clear the fraction.
• Be very cautious: the sign of the inequality may change depending on the value of the denominator. If the denominator could be negative, it may flip the inequality sign during solving.

In the original problem \( \frac{4}{y+2} <-2 \), the variable \( y \) is in the denominator, so we must multiply both sides by \( y+2 \) with caution. It's important to determine the cases where \( y+2 \) is positive or negative to ensure the correct inequality is preserved.

Solving Inequalities Step by Step

Solving inequalities requires a careful, step-by-step approach to ensure the solution is accurate and valid for all relevant cases. Here's how to tackle these problems:

• Isolate the variable term: Rearrange the inequality so that the term containing the variable is by itself on one side of the inequality.
• Clear fractions if any: Multiply through by the denominator to remove fractional terms, taking care with sign changes.
• Simplify the inequality: Combine like terms and simplify both sides wherever possible.
• Solve for the variable: Perform operations to solve for the variable, paying attention to any necessary sign changes, especially when multiplying or dividing by negative numbers.
• Check the solution: Ensure that the solution is valid for the domain of the variable (e.g., restrictions implied by zero denominators).

The original problem was solved methodically: from isolating the fraction to flipping the inequality sign when dividing by a negative, each step is key for correct solutions.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to make solving inequalities easier. Here’s how it works in this context:

• Rearranging Terms: Move terms from one side of the inequality to the other to isolate the variable term. This often involves addition or subtraction.
• Distributive Property: Use this property to expand expressions, such as multiplying a term by all parts of a bracketed expression.
• Combining Like Terms: Simplify expressions by adding or subtracting similar terms, which streamlines the inequality and focuses on the variable of interest.

In our exercise, algebraic manipulation helps transform \( 4 < -2(y+2) \) into \( 8 < -2y \) by distributing and combining like terms efficiently. Mastering these techniques is essential to solve inequalities accurately.