Problem 48 - Introductory Algebra for College Students

Question

Use the multiplication property of inequality to solve each inequality and graph the solution set on a number line. \(-7 x>21\)

Step-by-Step Solution

Answer

The solution to the inequality \(-7 x>21\) is \(x<-3\). This is represented on the number line as an open circle at -3, with a line extending to the left to illustrate that it includes all numbers less than -3.

Step 1: Setting up the inequality

To begin, the inequality \(-7 x>21\) is provided. The aim here is to isolate x on one side.

Step 2: Divide both sides by -7

Dividing both sides of the inequality by -7 gives \(x<-\frac{21}{7}\). Important to remember is that whenever we divide or multiply both sides of an inequality by a negative number, the direction of the inequality sign changes.

Step 3: Simplify the right side

Simplifying \(-\frac{21}{7}\) gives -3, therefore, the inequality \(x<-3\) is obtained.

Step 4: Solve for x

We now have the inequality \(x<-3\). This means that any number less than -3 is a solution to the inequality.

Step 5: Graph the solutions on a number line

On a number line, values less than -3 are represented by a line that starts at -3 and extends to the left. An open circle is drawn at -3 to show that -3 is not included in the solution set.

Key Concepts

Graphing Inequalities

Graphing inequalities on a number line is a way to visually represent the range of solutions for a given inequality. When dealing with an inequality such as \( x < -3 \), you are describing a range of values that satisfy this inequality—meaning any value of \( x \) less than -3 is a solution. On the number line, you start by locating -3, which acts as a boundary point. Since -3 is not part of the solution (because the inequality is strict, using the less than symbol <), you indicate this by drawing an open circle. This open circle shows that -3 is not included. The line then extends to the left, representing all numbers less than -3 being part of the solution set.
The visual representation helps to easily see which numbers satisfy the inequality, making it easier to understand and interpret the solution. When graphing, always remember that:

• An open circle is used for inequalities like < and >, indicating that the boundary value is not included.
• A closed circle is used for inequalities like \( \leq \) and \( \geq \), indicating that the boundary value is included.

Solving Inequalities

Solving inequalities involves finding all the possible values of a variable that satisfy the inequality condition. Just like solving equations, you want to isolate the variable on one side. However, inequalities require special attention because operations can affect the inequality's direction.
The multiplication property of inequality states that when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. This is crucial to remember, as illustrated in the exercise:

• Begin with the inequality \(-7x > 21\).
• To isolate \(x\), divide both sides by \(-7\), which flips the inequality direction, resulting in \(x < -3\).

Solving inequalities not only involves calculating values but understanding how operations affect the inequality itself. Always check the inequality direction, especially after multiplying or dividing by a negative, to ensure an accurate solution.

Algebraic Concepts

Algebraic concepts such as solving inequalities and understanding the properties of inequalities are fundamental in algebra. These concepts allow us to engage with mathematical operations to find solutions to problems and analyze different scenarios.
Key algebraic ideas in solving inequalities include:

• Properties of Inequalities: Knowing how operations (addition, subtraction, multiplication, division) interact with inequalities helps solve problems effectively.
• Direction of Inequality: Remember that inequalities can change direction when multiplied or divided by negative numbers, highlighting the importance of attending to operational signs.
• Simplification: Breaking down complex inequalities into simpler, equivalent forms makes finding solutions easier and clearer.

Understanding these algebraic principles not only helps in solving inequalities but also builds a broader foundation for dealing with more complex algebraic expressions in future learning.