Problem 123 - College Algebra

Question

Use the same technique employed in Exercises 121-122 to solve each inequality in Exercises 123-124. In each case, what conclusion can you draw? What happens if you try solving the inequalities algebraically? $$12 x-10>2(x-4)+10 x$$

Step-by-Step Solution

Answer

The inequality has no solution because -10 is not greater than -8.

Step 1: Distribute the terms

Start by distributing the 2 in 2(x-4) to get \(2x - 8\), now the inequality is \(12x - 10 > 2x - 8 + 10x\)

Step 2: Combine like terms

Combine all x terms and numerical terms separately on the right side of the inequality. The inequality becomes \(12x - 10 > 12x - 8\).

Step 3: Solve for x

Next, try to isolate x on one side of the inequality by subtracting 12x from both sides. The inequality becomes \(-10 > -8\).

Step 4: Analyze the inequality

The inequality -10 > -8 is incorrect, since -10 is not greater than -8. Thus, this inequality has no solution.

Key Concepts

Solving Inequalities

Solving inequalities is a process that closely resembles solving equations, but there is a critical difference. While the goal is still to isolate the variable, the presence of inequality signs such as \( >, <, \geq, \leq \) means we don't always end up with a single numeric solution. Instead, the solution may be a range of values that satisfy the inequality. Whenever we are solving inequalities:

• We treat the inequality sign like an equal sign when performing most algebraic manipulations.
• However, if we multiply or divide both sides of the inequality by a negative number, we must flip the inequality sign to maintain a true statement.

Taking these into account will ensure correct solutions in all steps of the manipulation process. In the example exercise, no solution is found because the inequality simplifies to a false statement, illustrating how certain inequalities can express contradictions.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve for variables. This process is crucial in solving inequalities, as it helps isolate the variable to understand its potential values. Here are some key manipulations typically employed:

• Adding or subtracting the same quantity from both sides keeps the inequality balanced.
• Combining like terms helps in simplifying the expressions involved.
• Isolating the variable often involves collecting all terms containing the variable on one side of the inequality.

For the inequality given in the original exercise, algebraic manipulation is used effectively. However, upon isolating the variable, the manipulation led to a statement \(-10 > -8\), which is not possible. This signifies no possible solution, emphasizing that contradictions can occur during correct algebraic manipulation.

Distributive Property

The distributive property is a foundational algebraic principle used to simplify expressions. It allows you to remove parentheses by "distributing" a factor across terms within the parentheses. This is key when attempting to solve equations or inequalities.In the context of solving the original inequality:

• The distributive property is applied to the term \(2(x-4)\), which simplifies to \(2x - 8\).
• This step is critical in ensuring that all terms involving the variable are clearly isolated, allowing for direct comparison and solution.

This property is always vital to ensure that each term is accurately represented and helps in laying out expressions correctly. Once properly distributed, the inequality resembles a standard form that can easily be manipulated algebraically.

No Solution Inequalities

Sometimes, an inequality can have no solution. This happens when the algebraic manipulations lead to a contradiction or a false statement. In the given exercise:

• After simplifying, the inequality reduces to \(-10 > -8\), which is not true.
• Such results indicate that there are no values satisfying the original inequality condition.

When faced with a no-solution inequality, it implies that there's no possible number that, when substituted into the original inequality, will make it true. This is a crucial realization in understanding inequalities because it reflects scenarios where conditions can never be met. Recognizing these outcomes strengthens comprehension of algebraic principles and the nature of inequalities.