Problem 123 - Introductory and Intermediate Algebra for College Students 4th

Question

$$\text { Solve: } x-2(3 x-2)>2 x-3$$

Step-by-Step Solution

Answer

The solution of the given inequality is \(x > -1/3\).

Step 1: Distribute

Apply the distributive property to both sides. This means to eliminate the parentheses by multiplying \(2\) by each term inside the parentheses. On the left side, \(2\) distributes over \(3x - 2\) to give \(x - 2*3x + 2*2 = 2x - 3\).

Step 2: Simplify

Combine like terms on both sides. On the left side, combine \(x\) and \(-6x\) to get \(-5x + 4 > 2x - 3\).

Step 3: Isolate the variable

To isolate \(x\), it is necessary to subtract \(2x\) from both sides and add \(3\) to both sides. Applying these operations yields: \(-5x + 2x < -3 + 4\).

Step 4: Solve the inequality

Combine like terms to get \(-3x < 1\). From here, divide by \(-3\) on both sides. Do not forget to invert the sign when dividing or multiplying an inequality by negative number, which gives \(x > -1/3\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to remove parentheses by distributing, or multiplying, a single term outside the parentheses across terms inside the parentheses. For example, if you have an expression like \( a(b + c) \), applying the distributive property means you calculate \( ab + ac \).
In solving inequalities, this step is crucial to simplify complex expressions. Like in our original problem, \( x - 2(3x - 2) > 2x - 3 \), the distributive property helps to simplify the expression \( -2 \times (3x - 2) \).

• First, multiply \(-2\) by \(3x\) to get \(-6x\).
• Then, multiply \(-2\) by \(-2\) to give \(+4\).

Thus, the left side of the inequality becomes \(x - 6x + 4\), setting the stage for the next steps.

Combining Like Terms

Combining like terms is the process of simplifying expressions by summing coefficients of terms that have the same variable.
This step is essential to make the equation or inequality easier to solve. In numeric terms, like combines with like. In terms of variables, terms like \(x\) and \(-6x\) combine because they share the variable \(x\).

• In our example, the terms \(x\) and \(-6x\) become \(-5x\).
• The constant term \(+4\) remains unchanged for now.

So, from our previous work, it results in the simplified inequality \(-5x + 4 > 2x - 3\). Simplifying expressions in this manner brings clarity and form, making further steps manageable.

Isolating Variables

Isolating the variable is a critical step in solving an inequality or equation, aiming to get the variable, usually \( x \), on one side of the inequality or equation by itself.
The goal is to find out what values the variable can take. In our problem, \(-5x + 4 > 2x - 3\), we want to move all terms involving \(x\) to one side. Here’s how it’s done:

• Subtract \(2x\) from both sides, leading to \(-5x - 2x + 4 > -3\).
• Add \(3\) to both sides to clean up the inequality, giving \(-7x > -7\).

Through such operations, you aim to handle the variable smartly and bring all necessary constants to the opposite side. Each operation adheres to algebraic rules ensuring the inequality's truth is preserved.

Inequality Properties

Understanding inequality properties is key because they guide us on how inequalities differ from equations. One crucial property to remember during manipulation, especially when dividing or multiplying both sides by a negative number, is the need to reverse the inequality sign.
In our final step, \(-3x < 1\), we needed to isolate \(x\) by dividing by \(-3\): this step is a perfect example of utilizing inequality properties effectively.

• When dividing by \(-3\), flip the inequality from \(<\) to \(>\).
• The correct result is \(x > -\frac{1}{3}\).

These properties ensure your solution is correct, reflecting the inequality’s behavior over its solutions. Thus, handling inequalities with these properties in mind helps achieve precise, accurate resolutions.