Problem 31 - College Algebra

Question

Solve each linear inequality in Exercises 27-48 and graph the solution set on a number line. Express the solution set using interval notation. $$-9 x \geq 36$$

Step-by-Step Solution

Answer

The solution to the inequality \( -9x \geq 36 \) is \( x \leq -4 \). This can be represented with interval notation as \( (- \infty, -4] \) and graphically with a closed circle at \( -4 \) and an arrow extending to the left.

Step 1: Solving the Inequality

Start by isolating \(x\) in the inequality \(-9x \geq 36\). To do this, divide both sides of the inequality by \(-9\). Remember to reverse the direction of the inequality sign when multiplying or dividing by a negative number. So \(-9x / -9 \leq 36 / -9 \) simplifies to \(x \leq -4\).

Step 2: Convert to Interval Notation

Now that we know \(x \leq -4\), we can express this as an interval \((- \infty, -4]\) - where the round bracket indicates that the number is not included and the square bracket indicates that the number is included.

Step 3: Graphing the Solution

To graph, draw a number line. Mark \( -4 \) on the number line. Since \( -4 \) is included in the solution, draw a closed circle at \( -4 \). Draw an arrow extending to the left from \( -4 \) to represent all numbers less than \( -4 \) in the solution set.

Key Concepts

Interval Notation

Interval notation is a method used to describe a set of numbers between given endpoints. This notation provides a clear and concise way to depict continuous subsets of real numbers. It uses brackets and parentheses to denote whether endpoints are included or excluded in the set.

• Square brackets \([ ]\) indicate the endpoint is included, also known as making the interval 'closed'.
• Parentheses \( ( ) \) mean the endpoint is not included, which indicates an 'open' interval.

For example, the inequality solution \((x \leq -4)\) is expressed in interval notation as \[(-\infty, -4]\].

Here, \(-\infty\) (negative infinity) uses a parenthesis because infinity is not a specific number and cannot be included. The \-4\ is enclosed with a square bracket, showing that \(-4\) itself is part of the solution set.

Number Line Graphing

Graphing inequalities on a number line is a visual way to represent a solution set and understand where numbers fit within the inequality. It allows students to see the span of solutions quickly and easily.

To graph \(x \leq -4\), follow these steps:

• Start by drawing a horizontal line, which will act as the number line.
• Mark the number \-4\ on this line.
• Since \-4\ is included in our solution, draw a closed circle (filled-in dot) at \-4\.
• Draw an arrow extending to the left from \-4\ to show that all numbers less than \-4\ are also solutions. The arrow indicates that the solution extends infinitely in that direction.

This graph effectively shows at a glance that any number on or to the left of \-4\ satisfies the inequality \((x \leq -4)\).

Inequality Solving

Solving linear inequalities involves finding all possible values for the variable that will make the inequality true. Unlike equations, inequalities tell us that one side is greater or lesser rather than equal.

To solve \-9x \geq 36:\:

• First, you need to isolate the variable on one side. Here, divide both sides by the coefficient of \-x\, which is \-9\.
• When dividing or multiplying an inequality by a negative number, always reverse the inequality sign. So \-9x / -9\ becomes \x\ and \36 / -9\ becomes \-4\, changing the inequality from \ \geq \ to \ \leq \.
• The solution from the calculation is \(x \leq -4\).

The inequality solution process shows how simple operations can uncover the range of solutions that satisfy the inequality condition.

College Algebra

College algebra involves a comprehensive exploration of algebraic concepts and their application, building a foundation necessary for advanced mathematics. In college algebra, understanding the principles of solving equations and inequalities is crucial.

Key aspects of college algebra include:

• Mastering arithmetic operations and their properties.
• Learning how to manipulate algebraic expressions and functions.
• Solving advanced equations and inequalities, as you saw with \(-9x \geq 36\).

In college algebra, solving inequalities like \(-9x \geq 36\) trains students in problem-solving and analytical thinking, critical for higher-level mathematics and various real-world applications. This discipline lays the groundwork for understanding more complex mathematical topics and developing skills for various fields.