Problem 38 - Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. See Example 5. $$ \frac{5}{16}(x+1) \geq \frac{1}{4}(x-3) $$

Step-by-Step Solution

Answer

The solution is \([-17, \infty)\).

Step 1: Expand the Inequality

First, we distribute the fractions on both sides of the inequality. On the left-hand side, distribute \( \frac{5}{16} \) to \( x + 1 \):\[\frac{5}{16}(x + 1) = \frac{5}{16}x + \frac{5}{16}\]On the right-hand side, distribute \( \frac{1}{4} \) to \( x - 3 \):\[\frac{1}{4}(x - 3) = \frac{1}{4}x - \frac{3}{4}\]Thus, the inequality becomes:\[\frac{5}{16}x + \frac{5}{16} \geq \frac{1}{4}x - \frac{3}{4}\]

Step 2: Eliminate the Fractions

To eliminate the fractions, find a common denominator for all fractions involved. Here, a common denominator is 16.Multiply every term in the inequality by 16 to clear the fractions:\[16 \left( \frac{5}{16}x + \frac{5}{16} \right) \geq 16 \left( \frac{1}{4}x - \frac{3}{4} \right)\]This simplifies the inequality to:\[5x + 5 \geq 4x - 12\]

Step 3: Solve the Inequality

Now, solve the inequality for \( x \).Subtract \( 4x \) from both sides:\[5x - 4x + 5 \geq -12\]This simplifies to:\[x + 5 \geq -12\]Subtract 5 from both sides to isolate \( x \):\[x \geq -17\]

Step 4: Graph the Solution on a Number Line

To graph \( x \geq -17 \), draw a number line, mark the point \( -17 \), and shade all points to the right, including \( -17 \), indicating that \( x \) can be any number greater than or equal to \( -17 \).Place a closed circle on \( -17 \) to show that \( -17 \) is included in the solution set.

Step 5: Write the Solution in Interval Notation

Express the solution \( x \geq -17 \) using interval notation.Since \( -17 \) is included in the solution, use a bracket:\[[-17, \infty)\]This interval represents all numbers starting from \( -17 \) to infinity, including \( -17 \) itself.

Key Concepts

Interval Notation

Interval notation is a concise way of expressing a set of numbers, typically those representing the solution to an inequality. It uses brackets and parentheses to show where a number set starts and ends.

In our original exercise, after solving the inequality, we found that the solution set for \( x \) is \( x \geq -17 \).

Here's how interval notation works:

• A square bracket "\([\)" means the endpoint is included in the set. Use this for inequality symbols like "greater than or equal to."
• A parenthesis "\(()\)" indicates the endpoint is not included, used for strict inequalities like "greater than."
• The interval notation for the infinity symbol, "\(\infty\)", always uses a parenthesis, because infinity is a concept, not a concrete number.

Therefore, \([-17, \infty)\) represents all numbers \(x\) that are \(-17\) or larger, where \(-17\) is part of the solution, while \(\infty\) suggests that there's no upper limit.

Common Denominator

When dealing with inequalities that include fractions, finding a common denominator is crucial to simplifying the expressions into a workable format. This method allows you to eliminate fractions altogether, making the inequality easier to solve.

The common denominator is essentially a shared multiple of the denominators within all the fractions of your equation.
In the provided exercise, our fractions had denominators of 16 and 4. Choosing 16 as the common denominator was logical because it is already one of the largest denominators. Here’s why finding a common denominator is important:

• By multiplying each term by the common denominator, we clear the fractions. This simplifies our inequality, transforming it into an equation with whole numbers.
• This technique helps prevent errors that might arise when working with fractional numbers.

Here is the simplification step: \[16(\frac{5}{16}x + \frac{5}{16}) \geq 16(\frac{1}{4}x - \frac{3}{4})\]which simplifies to: \[5x + 5 \geq 4x - 12\]This simplification removes the hassle of fractions, easing further calculations.

Graphing Solutions

Graphing the solutions of an inequality visually represents all possible solutions on a number line. This method helps in understanding the range of values that satisfy an inequality. In our step-by-step solution, we graphed the inequality \(x \geq -17\). Here’s how to graph inequalities effectively:

• First, draw a horizontal line, which symbolizes your number line.
• Identify the solution—here, \(-17\)—and mark it clearly on the number line.
• Since \(x\) can be equal to \(-17\), use a closed circle on \(-17\) to indicate inclusion.
• Shade the region to the right of \(-17\) because these are all numbers greater than \(-17\).

In this visual representation, any number on or to the right of \(-17\) falls into the solution set, confirming that it satisfies \(x \geq -17\). This graphical representation simplifies checking if specific numbers are within the solution range.