Problem 58 - Prealgebra

Question

Solve each inequality. $$12 r-4>7+12 r$$

Step-by-Step Solution

Answer

There is no solution as the inequality results in a false statement.

Step 1: Simplify both sides

Look at both sides of the inequality. The left-hand side is already simplified as \( 12r - 4 \), and the right-hand side is \( 7 + 12r \). Both are as simple as they can be at this stage.

Step 2: Eliminate the term with the variable from one side

Subtract \( 12r \) from both sides to eliminate the \( r \) term from one side. This gives: \( 12r - 4 - 12r > 7 + 12r - 12r \). Simplifying this, you get \( -4 > 7 \).

Step 3: Analyze the result

With the simplification, you get the inequality \( -4 > 7 \). This is clearly a false statement because \(-4\) is not greater than \(7\).

Step 4: Determine the solution set

Since \(-4 > 7\) is a false statement, it means there are no values of \( r \) that satisfy the inequality. Therefore, the solution to this inequality is no solution.

Key Concepts

Prealgebra

Prealgebra is like a bridge that prepares you for the world of algebra. It involves basic arithmetic operations and introduces you to concepts such as integers, fractions, decimals, and simple equations. When you begin exploring inequalities, you are starting to move beyond simple equations and into the realm of algebraic thinking.
One of the key skills in prealgebra is learning how to manipulate expressions to isolate variables or simplify problems. This involves understanding operations such as addition, subtraction, multiplication, and division, and applying properties like distributive, associative, and commutative laws.
By focusing on these basic concepts, you lay a solid foundation for understanding more complex algebraic concepts, including solving inequalities which require a firm grasp of these fundamental skills.

Solving Inequalities

When solving inequalities, you are finding the range of values that make an inequality true. Unlike regular equations, inequalities use symbols like \(>\), \(<\), \(\geq\), or \(\leq\) to express that one side is either larger or smaller than the other.
Here are steps you typically follow to solve inequalities:

• First, simplify both sides of the inequality by combining like terms or eliminating fractions.
• Next, isolate the variable on one side of the inequality. This might involve adding, subtracting, multiplying, or dividing terms across the inequality line.
• Always remember, if you multiply or divide by a negative number, you must flip the inequality sign to maintain the true relationship between the variables.
• Finally, evaluate whether the inequality has a solution or if an apparent statement is false like in our example \(-4 > 7\).

Dealing with inequalities involves careful manipulation and thoughtful analysis to ensure accuracy.

Mathematical Reasoning

Mathematical reasoning is the process of using logical thinking to come to a conclusion based on known facts and principles. In the context of solving inequalities, it is crucial for understanding the steps required to isolate the variable and interpret the results accurately.
Let’s break down the reasoning for solving the inequality from our example, \(12r - 4 > 7 + 12r\). After canceling the variable \(r\) on both sides, we ended up with \(-4 > 7\), which is a false statement. The reasoning here helps us conclude that no value of \(r\) can satisfy this inequality.
Through mathematical reasoning, you not only understand how to manipulate numbers and variables but also how to critically assess whether an outcome makes sense or requires further examination. Reasoning skills are essential for moving from prealgebra to more advanced mathematics, ensuring you not only perform calculations but also verify them logically.