Problem 19 - Algebra 2

Question

Solve each equation or inequality. Check your solutions. $$ \frac{10}{m+1}>5 $$

Step-by-Step Solution

Answer

The solutions are \( m < 1 \) and \( m \neq -1 \); in interval notation: \((-\infty, -1) \cup (-1, 1)\).

Step 1: Set up the Inequality

We start with the inequality \( \frac{10}{m+1} > 5 \). Our goal is to find the values of \( m \) that satisfy this inequality.

Step 2: Clear the Fraction

To eliminate the fraction, multiply both sides of the inequality by \( m+1 \), which is valid if \( m+1 eq 0 \). This gives us \( 10 > 5(m+1) \).

Step 3: Simplify the Expression

Expand the right side of the inequality: \( 10 > 5m + 5 \).

Step 4: Isolate the Variable

Subtract 5 from both sides to isolate the terms with \( m \): \( 5 > 5m \).

Step 5: Solve for m

Divide both sides by 5 to solve for \( m \): \( 1 > m \), or equivalently \( m < 1 \).

Step 6: Consider the Constraint

Since \( m+1 eq 0 \) is required to avoid division by zero, \( m \) cannot be \(-1\). Combine this with the solution \( m < 1 \).

Step 7: Solution Set

The solution set is \( m < 1 \) and \( m eq -1 \). In interval notation, this is \((-\infty, -1) \cup (-1, 1)\).

Step 8: Check the Solution

Select a value from the solution set, such as \( m = 0 \), and substitute back into the original inequality. Verify that \( \frac{10}{0+1} = 10 > 5 \). This confirms the solution is correct.

Key Concepts

Solving Inequalities

To solve inequalities means finding all values of a variable that make the inequality true. Let's start with the given inequality: \( \frac{10}{m+1} > 5 \). The goal is to manipulate this equation using algebraic operations so we can isolate the variable \( m \) on one side of the inequality.
Here's a guided approach:

• **Clear the Fraction:** Multiply both sides by \( m+1 \), ensuring \( m+1 eq 0 \) to keep the multiplication valid. This step eliminates fractions, making the equation easier to handle.
• **Simplify the Expression:** Distribute and rearrange terms if necessary. For this problem, you'll end up with an inequality where terms involving \( m \) are isolated.
• **Solve for the Variable:** Perform arithmetic operations, like adding, subtracting, or dividing, to isolate \( m \). Make sure to flip the inequality if you multiply or divide by a negative number, although that's not required in this case.

By following these steps, you can systematically find the range of values that \( m \) can take to satisfy the original inequality. Remember, these steps are general and can be applied to a variety of inequalities.

Interval Notation

Interval notation is a way of writing the solution set for inequalities. When determining where our solution lies for \( m < 1 \) and \( m eq -1 \), interval notation becomes quite handy.
The concepts here are as follows:

• **Use of Parentheses:** Parentheses \( ( \) or \( ) \) indicate that an endpoint is not included. For this inequality, \( m < 1 \) and \( m eq -1 \) mean that \( -1 \) and \( 1 \) are not included in the solution set.
• **Union:** Use \( \cup \) to join separate intervals. For solutions like \( m \) is less than \( 1 \) but not equal to \(-1\), the overall notation is a combination of two intervals: \(( -\infty, -1 ) \cup (-1, 1)\).

Writing the solution in interval notation makes it easier to see at a glance where the variable meets the inequality criteria, while also clearly showing where the variable values are excluded.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. In the inequality \( \frac{10}{m+1} > 5 \), each side represents an algebraic expression.
Learning how to manipulate these expressions is key to solving problems:

• **Operations on both Sides:** When you perform operations like multiplying both sides by \( m+1 \), you're simplifying an algebraic expression.This keeps the relationship between the expressions balanced.
• **Expanding Expressions:** For example, going from \( 5(m+1) \) to \( 5m + 5 \) is an expansion. This is crucial for simplifying complex expressions and solving inequalities.
• **Isolating Variables:** By keeping equations or inequalities balanced, you can slowly isolate the variable by performing the same operations to rearrange different parts of the expressions.

Understanding these concepts allows you to tackle much more complicated expressions as you advance in algebra, paving the way toward solving more challenging mathematical problems.