Problem 7 - Intermediate Algebra

Question

For Problems \(1-18\), solve each of the inequalities and express the solution sets in interval notation. $$ \frac{3-x}{6}+\frac{x+2}{7} \leq 1 $$

Step-by-Step Solution

Answer

The solution is \([-9, \infty)\).

Step 1: Eliminate Fractions by Finding a Common Denominator

The expression has denominators 6 and 7. The least common multiple of 6 and 7 is 42. Multiply every term in the inequality by 42 to eliminate the fractions:\[ 42 \left( \frac{3-x}{6} \right) + 42 \left( \frac{x+2}{7} \right) \leq 42 \times 1 \]This simplifies to:\[ 7(3-x) + 6(x+2) \leq 42 \]

Step 2: Distribute and Simplify

Distribute 7 into \((3-x)\) and 6 into \((x+2)\):\[ 7(3) - 7(x) + 6(x) + 12 \leq 42 \]This simplifies to:\[ 21 - 7x + 6x + 12 \leq 42 \]

Step 3: Combine Like Terms

Combine the like terms:\[ 21 + 12 - 7x + 6x \leq 42 \]This becomes:\[ 33 - x \leq 42 \]

Step 4: Isolate the Variable

Subtract 33 from both sides of the inequality to solve for \(x\):\[ -x \leq 9 \]To solve for \(x\), multiply both sides of the inequality by -1, remembering to reverse the inequality sign:\[ x \geq -9 \]

Step 5: Express the Solution in Interval Notation

The solution \(x \geq -9\) in interval notation is written as:\[ [-9, \infty) \]

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of the real number line. It’s a concise way to express the range of solutions for an inequality.
When solving inequalities like the given exercise, once you find the solution for the variable, you need to express it in interval notation.
Here’s how to understand it:

• For inclusive values (like \(x \geq -9\)), brackets \[ [ ] \] are used, meaning the endpoint is part of the solution set.
• For exclusive values (like \(x > x_0\)), parentheses \( ( ) \), meaning the endpoint is not included.
• The interval \( [a, b] \) includes all numbers from a to b.
• The notation can also indicate infinity (e.g., \( [-9, \infty)\) means that the solution set includes all numbers greater than or equal to -9 and extends indefinitely).

This notation provides a clear and standardized way to communicate solution sets for inequalities.
It avoids confusion, especially in cases where multiple parts of the interval are involved.

Common Denominators

Finding a common denominator is essential when working with fractions, especially in mathematical expressions like equations and inequalities.
This concept helps in eliminating fractions and simplifies calculations. Here’s how it works:

• A common denominator is a shared multiple of the denominators of the fractions involved.
• In the exercise, the fractions have denominators of 6 and 7. You find the least common multiple (LCM), which is 42 in this case.
• By multiplying each term in the inequality by 42, you can eliminate the fractions and simplify your problem to a basic algebraic expression.

Using a common denominator ensures that you work with whole numbers, making it easier to solve the inequalities.
It’s a crucial step that significantly simplifies the process of finding solutions, especially when dealing with complex inequalities.

Like Terms

Like terms are terms that contain the same variable raised to the same power.
Combining like terms is a critical algebraic technique used to simplify expressions and solve equations or inequalities. Here’s what you need to know:

• Terms are considered 'like' if they have identical variable parts (e.g., \(-7x\) and \(6x\) in our problem).
• When you combine like terms, you add or subtract their coefficients while keeping the variable part unchanged.
• In this exercise, combining \(-7x\) with \(6x\) simplifies the expression to \(-x\).

This makes the inequality easier to solve and more straightforward because you've reduced it to fewer terms.
Recognizing and combining like terms is a fundamental skill that helps streamline algebraic operations, allowing for efficient problem-solving.