Problem 52 - Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. $$ 5>\frac{7}{2} a-9 $$

Step-by-Step Solution

Answer

The solution is \(a < 4\), graph as an open circle at 4 with shading to the left, interval notation: \((-\infty, 4)\).

Step 1: Isolate the Variable Term

To solve the inequality \(5 > \frac{7}{2}a - 9\), we need to isolate the term with the variable \(a\). First, add 9 to both sides of the inequality to eliminate the constant on the right side:\[5 + 9 > \frac{7}{2}a - 9 + 9\]This simplifies to:\[14 > \frac{7}{2}a\]

Step 2: Solve for the Variable

Next, we need to solve for \(a\). To do this, multiply both sides by the reciprocal of \(\frac{7}{2}\) which is \(\frac{2}{7}\), to cancel out the fraction:\[14 \times \frac{2}{7} > a\]Calculate the left side:\[4 > a\]This can also be written as:\[a < 4\]

Step 3: Graph the Solution Set

To graph \(a < 4\) on the number line:1. Draw a number line.2. Locate the number 4 on the number line.3. Place an open circle at 4 to indicate that 4 is not included in the solution set.4. Shade the number line to the left of 4 to represent all numbers less than 4.

Step 4: Write the Solution in Interval Notation

In interval notation, the solution \(a < 4\) is written as:\[(-\infty, 4)\]The parenthesis indicates that 4 is not included in the solution set, and \(-\infty\) signifies that the interval extends indefinitely to the left.

Key Concepts

Inequality Notation

Inequality notation is a mathematical way of expressing the relationship between two values, where one value is not necessarily equal to another. In these expressions, inequality signs such as \(>\), \(<\), \(\geq\), and \(\leq\) are used, which stand for "greater than", "less than", "greater than or equal to", and "less than or equal to", respectively.

For example, in our exercise, the inequality \(5 > \frac{7}{2}a - 9\) uses the \(>\) symbol to indicate that the expression on the left is greater than the expression on the right. The goal when solving inequalities is to isolate the unknown variable on one side.

Here are some helpful tips for working with inequalities:

• Perform the same operation on both sides to maintain the balance of the inequality.
• If you multiply or divide both sides by a negative number, the inequality sign must be reversed.
• Always check your solution by substituting values to ensure they satisfy the original inequality.

Understanding these basics of inequality notation helps in solving the problem accurately.

Graphing Inequalities

Graphing inequalities visually represents the solution set of an inequality on a number line. It provides a clear picture of all the possible values that satisfy the inequality. In our exercise, the solution was found to be \(a < 4\).

Here's how to graph the inequality step by step:

• Start by drawing a horizontal number line. This will allow you to place your solution visually.
• On this number line, identify the critical point, which in our example is 4.
• Place an open circle around 4. An open circle is used because the inequality is strictly less than (<), meaning 4 is not included in the solution.
• Shade the line to the left of the circle to indicate all numbers less than 4 are included in the solution set.

Graphing provides an ease of understanding when dealing with inequalities, as it establishes a visual representation of the solution set.

Interval Notation

Interval notation is a concise way of describing sets of numbers along a number line, often used to express solutions to inequalities. It's especially helpful for communicating the range of possible values of a variable efficiently.

For our solution \(a < 4\), the interval notation is \((-\infty, 4)\). Here's a breakdown of how this works:

The interval notation \((-\infty, 4)\) tells us:

• The parenthesis \((-\infty\) signifies that the interval starts at negative infinity, indicating there is no lower bound.
• The comma separates the lower bound from the upper bound.
• The 4 is the upper bound of the interval.
• The parenthesis around 4 indicates that it is not included in the interval, consistent with the \(<\) in the inequality \(a < 4\).

Using interval notation simplifies the expression of solution sets and makes them easy to understand at a glance.