Problem 8 - Intermediate Algebra

Question

For Problems \(1-18\), solve each of the inequalities and express the solution sets in interval notation. $$ \frac{4-x}{5}+\frac{x+1}{6} \geq 2 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, -31]\).

Step 1: Simplify the Expression

First, combine the fractions on the left-hand side of the inequality. To do this, find the least common denominator (LCD) of 5 and 6, which is 30. Rewrite each fraction with the common denominator:\[\frac{4-x}{5} + \frac{x+1}{6} = \frac{6(4-x)}{30} + \frac{5(x+1)}{30}\]Combine and simplify:\[\frac{24 - 6x + 5x + 5}{30} = \frac{29 - x}{30}\]

Step 2: Solve the Inequality

Now substitute the simplified expression back into the inequality:\[\frac{29 - x}{30} \geq 2\]To eliminate the fraction, multiply both sides of the inequality by 30:\[29 - x \geq 60\]

Step 3: Isolate the Variable

Isolate \(x\) by subtracting 29 from both sides:\[-x \geq 31\]Next, multiply both sides by -1, remembering to reverse the inequality sign:\[x \leq -31\]

Step 4: Express the Solution in Interval Notation

The solution to the inequality is all \(x\) values less than or equal to -31. In interval notation, this is expressed as:\[(-\infty, -31]\]

Key Concepts

Interval Notation

Interval notation is a shorthand used to describe a range of numbers in mathematics. Instead of listing all possible numbers that can be solutions, we use a concise format that indicates the starting and ending points of the range. This format is particularly useful in inequalities.
Understanding the symbols:

• ( or ) are used for open intervals, meaning the endpoint is not included in the solution set.
• [ or ] are used for closed intervals, meaning the endpoint is included.

For example, the expression \((-\infty, -31]\) includes all numbers less than or equal to \(-31\), but not infinity, as it cannot be reached. The \(-\infty\) is always paired with a parenthesis since infinity and negative infinity are concepts, not numbers. In interval notation, our solution \((-\infty, -31]\) accurately captures the essence of the inequality solution.

Algebraic Expressions

Algebraic expressions are fundamental to solving inequalities like the one given in the exercise. They consist of variables, numbers, and operations (like addition, subtraction, multiplication, and division).
In the exercise, the algebraic expression was initially \(\frac{4-x}{5}+\frac{x+1}{6}\geq 2\). The goal is to simplify this expression so that we can solve the inequality more easily. Simplification typically involves operations like:

• Finding a common denominator to combine fractions, as was done with \((4-x)\) and \(x+1\).
• Rewriting fractions with the least common denominator (LCD), which in this case was 30.
• Simplifying the expression by combining like terms, which leads to a more manageable inequality.

By understanding and manipulating algebraic expressions, we can effectively reduce complex problems into simpler statements that are easier to solve.

Inequality Solutions

Solving inequalities involves finding all the values of the variable that make the inequality true. The process can be broken down into clear steps:

• Simplify the expression: This involves combining like terms and reducing complex expressions into manageable lines, as shown with finding the LCD and rewriting the fractions.
• Eliminate fractions: Multiply through by the denominator to clear fractions and work with whole numbers, as demonstrated when multiplying by 30 in the problem.
• Isolate the variable: Rearrange the inequality to get the variable alone on one side, adjusting the inequality sign if you multiply or divide by a negative number.
• Express the solution: Finally, express the variable's range using interval notation, indicating clearly whether endpoints are included or not.

For example, isolating \(x\) and solving results in \(x \leq -31\), giving us a solution set in interval notation of \((-\infty, -31]\). This step-wise method ensures clarity and accuracy for solving inequalities.