Problem 81 - Introductory and Intermediate Algebra for College Students 4th

Question

Solve each inequality. $$4 x-4<4(x-5)$$

Step-by-Step Solution

Answer

The inequality has no solution, as the final statement is not valid.

Step 1: Distribute

First distribute the 4 on the right-hand side of the inequality. This results in \(4x-4<4x-20\).

Step 2: Simplify

Next, subtract \(4x\) from both sides of the inequality. Please do note that when manipulating the inequality with addition or subtraction, the sign of inequality remains the same. After subtracting \(4x\), the inequality simplifies to: \(-4<-20\).

Step 3: Check Validity

At this point, we have a statement which reads: '-4 is less than -20'. This is not true, therefore there is no solution for this inequality.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle that helps you simplify expressions and solve equations or inequalities that involve parentheses. It states that when you multiply a single number or variable by a sum or difference inside parentheses, you should distribute the multiplication across each term within the parentheses. For example:

• When you have an expression like \(a(b + c)\), you distribute \(a\) to both \(b\) and \(c\), resulting in \(ab + ac\).
• In the given inequality \(4x - 4 < 4(x - 5)\), the distribution involves multiplying \(4\) by both \(x\) and \(-5\). This gives us \(4x - 20\).

Recognizing and using the distributive property correctly allows you to break down and simplify more complex algebraic expressions, setting the stage for solving equations and inequalities.

Simplifying Expressions

Simplifying expressions is essential in mathematics, as it allows complicated expressions to be expressed more clearly and managed more easily. Once you use the distributive property, the next step is usually to simplify the expressions by combining like terms and performing arithmetic operations where possible.
In our example \(4x - 4 < 4x - 20\), simplifying involved:

• Eliminating the \(4x\) on both sides: Subtracting \(4x\) from both sides yields \(-4 < -20\).

This doesn't mean our work is done; it’s just a necessary step to further progress towards understanding the solution or lack thereof. Simplifying helps make problems more understandable and paves the way for more extensive algebraic manipulation.

No Solution

At times, you will find that an inequality simplifies to a statement that is not logically or mathematically possible, like saying a smaller number is less than a larger number when it's not. In our problem, we ended up with:

• \(-4 < -20\)

This statement is false because \(-4\) is not less than \(-20\). When you encounter an impossible statement during problem solving, it indicates that the original inequality has no solution. There's nothing you can substitute for your variable \(x\) that will make the inequality true.
"No solution" might seem disappointing, but it's a valid outcome in algebra, demonstrating the importance of evaluating the expressions correctly at each step.

Algebraic Manipulations

Algebraic manipulations involve making strategic changes to equations or inequalities without altering their fundamental relationships. These manipulations include adding, subtracting, multiplying, or dividing both sides of an equation or inequality by the same value.
For inequalities:

• When you add or subtract the same number from each side, the inequality's direction stays the same.
• Pay attention during multiplication or division by a negative number, as this operation reverses the inequality sign.
• Always aim to isolate the variable to determine its permissible values.

In our exercise, subtracting \(4x\) from both sides was critical in simplifying the inequality to reach an evaluation. Mastering algebraic manipulations helps in achieving the solution efficiently and correctly.Continued practice helps to develop a keen eye for the right manipulations necessary for each problem encountered.