Problem 82 - Introductory Algebra for College Students

Question

Solve each inequality. \(3 x-5<3(x-2)\)

Step-by-Step Solution

Answer

The inequality has no solutions.

Step 1: Distribute the multiplication

Rewrite \(3(x-2)\) by distributing the multiplication: \(3 * x - 3 * 2 = 3x - 6\). So the inequality becomes \(3x - 5 < 3x - 6\).

Step 2: Eliminate similar terms

The inequality \(3x - 5 < 3x - 6\) has 3x on both sides. Subtract 3x from both sides to cancel out this term. This results in \(-5 < -6\).

Step 3: Checking the result

The inequality \(-5 < -6\) is not true. This means the original inequality has no solutions, because no value of x would satisfy the inequality.

Key Concepts

Algebraic Inequalities

When it comes to understanding algebraic inequalities, you're dealing with mathematical expressions that show the relative size or order of two values. They are not much different from the equalities you're used to in algebra—except, rather than showing how two quantities are the same, they illustrate how one is larger or smaller than the other. Inequalities use signs such as < (less than), <= (less than or equal to), > (greater than), and >= (greater than or equal to).

For instance, in the exercise \(3x - 5 < 3(x - 2)\), we are looking for all the values of \(x\) that make the statement true. The process involves treating the inequality similarly to an equation by performing operations to isolate \(x\), but with extra caution: if you multiply or divide by a negative number, the inequality sign must be flipped (reversed). However, this particular example doesn't entail such complexity; instead, as we'll see, other properties of inequalities come into play.

Inequality Properties

Understanding the properties of inequalities is crucial for solving them correctly. Here are a few basics to keep in mind:

• The addition or subtraction property: You can add or subtract the same number from both sides of the inequality without changing its direction.
• The multiplication or division property: Multiplying or dividing both sides by a positive number maintains the direction of the inequality. If the number is negative, the direction of the inequality reverses.
• The transitive property: If \(a > b\) and \(b > c\), then \(a > c\).
• The division property: Inequalities remain true when both sides are divided by the same positive number.

Notably, the step-by-step solution provided doesn't require flipping the inequality sign because no multiplication or division by negative numbers is involved. However, the properties are essential when simplifying the inequality during the problem-solving process.

Distributive Property

The distributive property is a cornerstone of algebra that allows us to simplify expressions and solve equations and inequalities. It states that you can distribute multiplication over addition or subtraction within parentheses. Mathematically, this is shown as \(a(b + c) = ab + ac\) or \(a(b - c) = ab - ac\).

Looking at our example, the exercise started with applying the distributive property: \(3(x - 2)\) becomes \(3*x - 3*2\), which simplifies to \(3x - 6\). This step is foundational because it transforms the inequality into a form where like terms can be collected and compared, leading to a clearer understanding of the solution. While the distributive property made the inequality simply to solve, it's important to remember that recognizing when and how to apply it is a key skill in algebra.