Problem 51 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

\(x > \frac{4}{3}\)

Step 1: Rearrange the Inequality

First, let's get rid of the fraction by multiplying both sides by the denominator \(x - 3\), assuming \(x eq 3\) since that would make the denominator zero.\[ (x + 2) > -2(x - 3) \]

Step 2: Simplify the Expression

Expand and simplify the inequality from Step 1.\[ x + 2 > -2x + 6 \]

Step 3: Solve for x

Now, solve the inequality for \(x\) by first adding \(2x\) to both sides:\[ 3x + 2 > 6 \]Then, subtract 2 from both sides:\[ 3x > 4 \]Finally, divide both sides by 3:\[ x > \frac{4}{3} \]

Step 4: Consider the Domain

Recall that \(x eq 3\) because it would make the original expression undefined. However, since \(x > \frac{4}{3}\), there is no conflict with \(x = 3\). The solution is consistent with this restriction on \(x\).

Key Concepts

Inequality Manipulation

In solving inequalities, it's important to understand that inequalities behave similarly to equations when solving for a variable, but with some additional rules.
When you multiply or divide both sides of an inequality by a negative number, the direction of the inequality changes. This is crucial to remember as it can affect the solution set significantly.
In our exercise, we started by multiplying both sides of the inequality by the denominator, \(x - 3\), which is permitted here because \(x eq 3\). This approach helps eliminate the fraction, making the inequality easier to solve.

• Distributing and combining like terms can help simplify complex inequalities.
• Always remember to check for changes in the inequality sign, especially during multiplication or division by negatives.
• Simplifying and reducing terms as much as possible is key to finding the correct solution.

It's all about maintaining the balance and logic of the inequality as you work through it.

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are both polynomials. They require careful handling due to their potential to become undefined when the denominator is zero.
In the exercise above, the original inequality involves a rational expression, \( \rac{x+2}{x-3} \gt -2 \). It's essential to manage rational expressions carefully, keeping in mind the points where they might not be defined.

• Identify if any values of the variable will make the denominator zero as these are usually restrictions in the domain.
• When dealing with rational expressions, it's common to multiply both sides by the denominator to clear it, making sure not to include the values that would lead to division by zero.
• This process helps transform a rational inequality into a simpler and non-rational inequality, which can be solved using similar methods to solving linear equations.

By understanding these properties, you can carefully and accurately work with inequalities involving rational expressions.

Domain Restrictions

Domain restrictions are conditions that restrict the values the variable can take without causing mathematical problems like division by zero.
In our given problem, the rational expression \( \rac{x+2}{x-3} \) has an inherent domain restriction due to the denominator \(x - 3 \). Since we cannot divide by zero, \(x = 3\) is not part of the domain.
While solving, we find that \(x > \rac{4}{3}\) is the solution, thus avoiding the restriction naturally.

• When solving inequalities involving rational expressions, always first identify these restrictions.
• Ensure that the final solution respects these restrictions by excluding those restricted values.
• Consider the implications of these restrictions throughout the process to avoid incorrect solutions that seem mathematically valid but are actually not possible due to the domain limitations.

Remembering the impact of domain restrictions helps avoid potential errors and ensures that the solution is valid for all defined values of the variable.