Problem 28 - College Algebra and Calculus: An Applied Approach

Question

Copy and complete the statement using the correct inequality symbol. If \(5-3 x>-7\), then \(x\) _______4.

Step-by-Step Solution

Answer

The value of \(x\) is less than 4: \(x < 4\).

Step 1: Rearrange the inequality

To isolate \(x\), start with the given inequality \(5-3x>-7\). The goal is to have \(x\) on one side of the equation. To do this, subtract 5 from both sides of the inequality to get \(-3x > -7 - 5\). This simplifies to \(-3x > -12\).

Step 2: Divide by -3

Next, divide every term by -3 (the coefficient of \(x\)) to get \(x < 4\). Important to note, when you divide or multiply an inequality by a negative number, you have to switch the direction of the inequality sign.

Step 3: Interpret the result

The inequality \(x < 4\) means that the value of \(x\) is less than 4.

Key Concepts

Understanding Algebra

Algebra is an essential branch of mathematics that involves working with symbols and the rules for manipulating these symbols. In algebra, we often work with variables, which are symbols that represent numbers. Solving an algebraic expression means finding the value(s) of the variable(s) that make the equation true.
In the exercise we've seen, algebra helps us manipulate the expression given in the inequality. We start by isolating the variable, which is achieved by performing operations on both sides of the inequality. This process doesn't change the fundamental truth of the inequality but helps clarify the relationship between the variables involved.
Whenever you're dealing with an equation or an inequality, think of it like a balance scale. Each side of the equation has a weight, and you need to perform the same operation on both sides to keep the balance. This way of thinking is crucial for anyone learning to solve equations and inequalities in algebra.

Inequality Symbols and Their Meaning

Inequality symbols are used in mathematics to show the relationship between two values. Unlike equations, which indicate that two values are equal, inequalities suggest that one value is not equal to the other. Here are some common inequality symbols you might encounter:

• \(>\) : greater than
• \(<\) : less than
• \(\geq\) : greater than or equal to
• \(\leq\) : less than or equal to

In the given exercise, the symbol \(>\) is initially used to indicate that the expression \(5 - 3x\) is greater than -7. By rearranging and solving the inequality, we determine that \(x < 4\). When solving inequalities, it's crucial to remember that multiplying or dividing both sides by a negative number reverses the inequality symbol. This rule ensures the inequality expresses the correct relationship between the variables.

Applying Mathematical Reasoning

Mathematical reasoning is the process of using logical thinking to solve problems and make sense of numbers, symbols, and relationships. It's about identifying patterns, making conjectures, and proving them if possible. In our exercise, reasoning plays an important role in understanding and manipulating the inequality to find a solution.
When tackling inequalities, each step must be considered carefully. For example, isolating the variable involves logical steps such as addition, subtraction, multiplication, or division. Each operation needs to maintain the truth of the original inequality.
It's important to remember that inequalities do not always have a single solution. Instead, they describe a range of possible values that satisfy the condition. When interpreted mathematically, the solution \(x < 4\) tells us that any number less than 4 will satisfy the inequality \(5 - 3x > -7\). This analytical approach requires a good grasp of both algebraic manipulation and the principles of inequalities.