Problem 57 - Introductory and Intermediate Algebra for College Students 4th

Question

Use both the addition and multiplication properties of inequality to solve each inequality and graph the solution set on a number line. $$2 x-3>7$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x > 5\), which represents all numbers greater than 5.

Step 1: Isolate the term containing variable

Begin by adding 3 to both sides of the inequality to isolate the term with the variable on the left side. This gives \(2x > 7 + 3\) which simplifies to \(2x > 10\).

Step 2: Isolate the variable

Next, divide both sides of inequality by 2 to solve for x. This results in \(x > 5\).

Step 3: Graph the solution on a number line

Draw a number line and indicate 5 with an open circle (as 5 is not included in the solution). Shade to the right of 5 to represent all numbers greater than 5.

Key Concepts

Inequality Properties

Understanding the properties of inequalities is crucial when working with algebraic expressions and equations. An inequality is a mathematical sentence indicating that two expressions are not equal, and it involves symbols such as > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to).

One key property to remember is the Addition Property of Inequality, which states that you can add or subtract the same number from both sides of the inequality without changing its direction. For instance, if we have an inequality like \( 2x - 3 > 7 \), we can add 3 to both sides to keep the inequality in balance, resulting in \( 2x > 10 \). This step simplifies the inequality and brings us closer to finding the solution for the variable.

The Multiplication Property of Inequality is another fundamental concept, which indicates that we can multiply or divide both sides of the inequality by the same positive number without altering the direction of the inequality. However, if we multiply or divide by a negative number, we must reverse the direction of the inequality. For example, in the aforementioned case, dividing both sides by 2 gives us \( x > 5 \). It's essential to be attentive to these properties because they are vital for solving inequalities correctly.

Number Line Graphing

Graphing the solution of an inequality on a number line is a visual way to represent the range of values that satisfy the inequality. It can be incredibly helpful in understanding and interpreting the solution set. For an inequality like \( x > 5 \), we can illustrate this on a number line.

To graph the solution:\( x > 5 \), we place an open circle at the number 5 on the number line to indicate that the value of 5 is not included in the solution set—hence the term 'open' circle. We then shade the area to the right of 5 to represent all values that are greater than 5, essentially creating a visual representation of the solution set.

Why We Use Open and Closed Circles

Open and closed circles are used to graphically denote whether a number is included in the solution set. Open circles signify 'not included' (for > and <), while closed circles mean 'included' (for ≥ and ≤). It's a simple yet powerful tool for expressing ranges and helps in verifying whether a particular value belongs to the solution set or not.

Isolation of Variables

Isolation of the variable is a fundamental step in solving both equations and inequalities. The goal is to get the variable by itself on one side of the equation or inequality, which makes the value of the variable immediately clear. For inequalities, following the correct properties is crucial in maintaining the inequality's integrity while isolating the variable.

The process involves using inverse operations to systematically undo any addition, subtraction, multiplication, or division that is performed on the variable. Taking our previous example \( 2x > 10 \), the variable x is isolated by dividing both sides of the inequality by 2. As a result, we get \( x > 5 \), which reveals the set of all possible values for x that make the original inequality true.

Always Check Your Solution

Although the steps seem straightforward, one valuable piece of advice when isolating the variable is to always check your solution. Substituting the found solution back into the original inequality can confirm whether the variable has been isolated correctly and if the solution is indeed valid.