Problem 53 - Elementary and Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it in interval notation. $$ 9 a+11 \leq 29 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, 2]\). Graph: all numbers to the left of and including 2.

Step 1: Understand the Inequality

The given inequality is \( 9a + 11 \leq 29 \). Our goal is to find all values of \( a \) that satisfy this inequality.

Step 2: Isolate the Variable Term

To isolate \( 9a \), subtract 11 from both sides of the inequality: \[9a + 11 - 11 \leq 29 - 11\]This simplifies to \[9a \leq 18\]

Step 3: Solve for the Variable

To solve for \( a \), divide both sides of the inequality by 9:\[\frac{9a}{9} \leq \frac{18}{9}\]This simplifies to \[a \leq 2\]

Step 4: Graph the Solution Set on a Number Line

On a number line, shade all the numbers from negative infinity to 2. Place a closed circle at 2 to indicate that 2 is included in the solution set since the inequality is \( \leq \).

Step 5: Write the Solution in Interval Notation

In interval notation, the solution set is written as\[(-\infty, 2]\]This indicates all real numbers less than or equal to 2 are part of the solution.

Key Concepts

Graphing Inequalities

Graphing inequalities is a straightforward way to visually represent all possible solutions to an inequality. In this exercise, we deal with the inequality \(a \leq 2\). Let's break it down into simple steps:

• Draw a number line that includes the number 2.
• Use a closed circle on 2 to indicate that this number is part of the solution set because the inequality uses \(\leq\).
• Shade the line extending to the left of 2, continuing towards negative infinity, because all numbers less than or equal to 2 satisfy the inequality.

This shaded region visually shows that every number on the left of 2 and including 2 itself is a solution to the inequality. Graphing helps not just to solve inequalities but also to understand their scope and range.

Interval Notation

Interval notation is a concise method for expressing a set of numbers along a number line. It conveys all solution values for an inequality in a neat format without needing to use words or additional symbols. In our solution:

• The inequality \(a \leq 2\) is expressed in interval notation as \((-\infty, 2]\).
• The round parenthesis \((-\infty\) indicates that we approach negative infinity, which can never be reached. That's why it's not included in the solutions.
• The square bracket \([2\) shows that the number 2 is included in the solution set because 2 is a value that satisfies the inequality.

Remember, using interval notation gives you a cleaner and faster way to indicate the set of solutions, which is especially useful when dealing with complex systems of inequalities.

Algebraic Solution Steps

Solving inequalities algebraically involves logical steps similar to solving equations, but with particular attention to the direction of the inequality symbol. Here's a breakdown of the steps:

• Understand the given inequality. In this case, \(9a + 11 \leq 29\).
• Isolate the term with the variable. Subtract 11 from both sides to get \(9a \leq 18\).
• Solve for the variable. Divide each side by 9, resulting in \(a \leq 2\).

With inequalities, it’s important to remember that when multiplying or dividing by a negative number, the inequality sign flips direction. However, as shown, that doesn't apply here since we're dividing by a positive number, 9. Always check your final inequality by substituting back into the original equation to ensure it holds true.