Problem 83 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each inequality. $$x+3

Step-by-Step Solution

Answer

The solution to this inequality is that it is always true, there is no specific range for x.

Step 1: Isolate x term

Subtract 'x' from both sides to get the equation in standard form: \(x + 3 - x < x + 7 - x\). Simplifying this gives \(3<7\).

Step 2: Check if the inequality is true

Since 3 is less than 7, the inequality is always true.

Key Concepts

Isolating Variable

When solving inequalities, isolating the variable is often the first step. This means you want the variable to be on one side of the inequality symbol,
while everything else is on the other side. In the given exercise, you might be tempted to solve for the variable 'x'. However, you quickly find that 'x' cancels itself on both sides. To isolate a variable, perform the same arithmetic operation on both sides, such as:

• Add or subtract the same number from both sides.
• Multiply or divide both sides by the same nonzero number.

The goal is to simplify the inequality or equation to one where the variable is by itself on one side. In our exercise, subtracting 'x' from both sides leads us to see that the inequality actually simplifies to a true statement about numbers, confirming our understanding of the problem.

Standard Form of Inequality

The standard form of an inequality has the variable terms listed first followed by the inequality symbol, ending with any constant terms.
This arrangement makes it easier to compare the size of expressions and understand what the inequality is stating. By subtracting 'x' from both sides in the original inequality \( x + 3 < x + 7 \), it simplifies to \( 3 < 7 \) - a straightforward inequality.This transformation shows the true nature of the problem: the original inequality expresses a truth of numbers, without an insolvable variable component. Therefore, when confronted with an inequality involving variables, your task is to maneuver it into this standard form step by step, giving clearer insight into what the inequality truly illustrates.

Simplifying Inequalities

Simplifying inequalities may sometimes seem redundant, but it is a crucial step in confirming your results.
Once an inequality is simplified into an evident true or false statement, it represents the set of values over which the original inequality holds.In our case, the inequality simplified to \( 3 < 7 \), which is always true. This tells us that the original inequality had no restrictions based on 'x'.Simplification involves a number of techniques:

• Use basic arithmetic operations to reduce terms.
• Cancel out terms on both sides of the inequality.
• Combine like terms to simplify expressions.

Such steps allow us to decipher complex inequalities into more digestible forms, aiding in better understanding and effective problem-solving skills. This is an essential tool for students working through algebraic inequalities.