Problem 53 - Beginning and Intermediate Algebra

Question

Solve each rational inequality. Graph the solution set and write the solution in interval notation. $$\frac{3}{a+7} \geq 1$$

Step-by-Step Solution

Answer

The solution to the rational inequality \(\frac{3}{a+7} \geq 1\) is \(a \leq -4\). The solution set in interval notation is \((-\infty, -4]\).

Step 1: Remove the fraction by finding a common denominator

To eliminate the fraction, we can multiply both sides of the inequality by the denominator of the fraction, which is (a+7), as long as (a+7)≠0: \((a+7) \cdot \frac{3}{a+7} \geq (a+7) \cdot 1\) Simplify the inequality to get the expression: 3 \(\geq a+7\)

Step 2: Solve the inequality for 'a'

Now we need to solve the inequality for the variable 'a'. To do this, we will subtract 7 from both sides of the inequality: 3 - 7 \(\geq a + 7 - 7\) Simplifying the inequality gives: -4 \(\geq a\) To write this inequality using the variable 'a' on the left side: a \(\leq -4\)

Step 3: Graph the solution set

To graph the solution set on a number line, draw an arrow extending to the left from -4 and place a closed circle at -4 to indicate that the solution includes -4: <---(-4)-(---->

Step 4: Write the solution in interval notation

The solution set on the interval is all real numbers less than or equal to -4. We express this in interval notation as: \[(-\infty, -4]\] So, the solution to the rational inequality, \(\frac{3}{a+7} \geq 1\), is \(a \leq -4\), and the solution set in interval notation is \((-\infty, -4]\).

Key Concepts

Graphing Solutions

Graphing solutions to inequalities involves using a number line to visually represent the range of values that solve the inequality.
This graphical representation helps to visualize the solution and see if it involves a continuous range or specific points. In the given exercise, solving the inequality gives us \( a \leq -4 \).

Here's how we graph it:

• First, locate the number -4 on a horizontal number line.
• Next, because the inequality is \( a \leq -4 \), include -4 in the solution by marking it with a closed circle.
• Draw a line or arrow extending to the left from the circle, indicating that all numbers to the left of -4 satisfy the inequality.

This graph represents all values of \( a \) less than or equal to -4.

Interval Notation

Interval notation is a way of writing subsets of the real number line. It is a concise form to represent continuous sets and is often used with inequalities.
In the language of intervals, we describe a solution set without using any words.

For the inequality \( a \leq -4 \):

• The solution includes every real number less than or equal to -4.
• We denote it as \((-\infty, -4]\).

Using interval notation, \( -\infty \) denotes that there is no lower bound. The parenthesis \((\) with \( -\infty \) signifies that \(-\infty\) is never included in the set. The bracket \([\) with -4 shows that this endpoint is included in the solution.
Hence, \((-\infty, -4]\) includes all real numbers from negative infinity up to, and including, -4.

Solving Inequalities

Solving inequalities often involves finding values that satisfy a mathematical statement involving less than, greater than, or equal signs. Rational inequalities, like the one in this exercise, require careful manipulation of fractions.

Let's break down the steps:

• First, eliminate fractions by multiplying each side by a common denominator, ensuring it is not zero.
• Then, isolate the variable by performing algebraic operations, like adding, subtracting, multiplying, or dividing both sides by the same number.
• Remember, if you multiply or divide both sides by a negative number, the inequality sign must flip.

In our example, after handling the fraction and simplifying, we obtained \( 3 \geq a+7 \). Further solving gives \( a \leq -4 \).
This process helps identify the set of values that make the original inequality true.