Problem 53 - Algebra A Combined Function

Question

Solve each inequality. Write each answer using solution set notation. $$ -5(1-x)+x \leq-(6-2 x)+6 $$

Step-by-Step Solution

Answer

The solution is \( \{ x \mid x \leq \frac{5}{4} \} \).

Step 1: Distribute and Simplify

First, we need to eliminate parentheses by distributing the numbers outside the parentheses across the terms inside. Start with the left side, \(-5(1-x) + x\), which becomes \(-5 + 5x + x\), simplifying to \(5x + x - 5\) or \(6x - 5\). On the right side, \(-(6 - 2x) + 6\) simplifies to \(-6 + 2x + 6\), which simplifies further to \(2x\). Now our inequality is \(6x - 5 \leq 2x\).

Step 2: Isolate Variable Term

Next, we need to isolate the variable term on one side of the inequality. We can do this by subtracting \(2x\) from both sides: \(6x - 5 - 2x \leq 2x - 2x\), which simplifies to \(4x - 5 \leq 0\).

Step 3: Solve for x

Now that the inequality is simplified to \(4x - 5 \leq 0\), we need to solve for \(x\). Add \(5\) to both sides to get \(4x \leq 5\). Now, divide both sides by \(4\) to isolate \(x\): \(x \leq \frac{5}{4}\).

Step 4: Write Solution in Set Notation

Now that we've solved the inequality, let's express it in solution set notation. Since \(x\) can be less than or equal to \(\frac{5}{4}\), the solution set is \( \left\{ x \mid x \leq \frac{5}{4} \right\} \).

Key Concepts

Solution Set Notation

Solution set notation is a systematic and concise way of expressing the solutions to inequalities and equations. This notation allows us to neatly represent all the values that satisfy a given inequality.
When we solve an inequality like the one in the exercise, we determine which values of the variable satisfy the inequality, and then express those values in set notation.
For instance, our solution was that the inequality had solutions where the variable is less than or equal to \( \frac{5}{4} \). Therefore, in set notation, we write this as:

• \( \{ x \mid x \leq \frac{5}{4} \} \)

The "\( \mid \)" symbol means "such that," and the set expresses all numbers \( x \) that are less than or equal to \( \frac{5}{4} \).
This notation is especially helpful because it is universal—mathematicians and students alike recognize and understand it. It also provides a clean way to present infinite solutions, such as all values less than \( \frac{5}{4} \).
Using solution set notation can simplify the presentation of your final answer and makes it clear for anyone reading your work.

Distributive Property

The distributive property is one of the essential rules for working with equations and inequalities. It helps you remove parentheses and distribute a multiplication over addition or subtraction.

In the exercise, we used the distributive property to expand both sides of the inequality:

• On the left side: \(-5(1 - x) + x\) becomes \(-5 + 5x + x\).
• On the right side: \(-(6 - 2x) + 6\) simplifies to \(-6 + 2x + 6\).

When using the distributive property, always multiply each term inside the parentheses by the term outside the parentheses. This rule applies for both positive and negative coefficients.
The result of applying this property is an expression without parentheses, which is often easier to simplify and solve. Remember, if you're dealing with subtraction, take care to correctly distribute that operation across each term inside the parentheses.

Isolate Variable

Isolating the variable is a vital step in solving equations or inequalities. This process involves gathering all the variable terms on one side and constant terms on the other to solve for the variable.
In the step-by-step solution, after simplifying both sides using the distributive property, we aimed to isolate our variable \(x\) as follows:

• Subtract \(2x\) from each side to gather \(x\) terms: \(6x - 2x - 5 \leq 2x - 2x\).
• This simplifies to \(4x - 5 \leq 0\).

By isolating \(x\), you aim to achieve a statement where the variable is the subject of the inequality or equation, making it straightforward to further solve for \(x\).
Once isolated, you can handle the inequality or equation using basic arithmetic operations, allowing you to easily determine the solution set.

Simplifying Expressions

Simplifying expressions is an important part of solving equations and inequalities. The aim is to make expressions as simple and as clear as possible, which can simplify further steps in problem-solving.
In our worked exercise, we simplify expressions through a series of steps:

• Apply the distributive property to get rid of parentheses.
• Combine like terms, such as \(5x + x = 6x\), to simplify the expressions.
• Eliminate any constant terms to finish simplifying the expression.

By simplifying expressions correctly, you make the problem easier to understand and solve. Simplified expressions often reveal clearer relationships between terms, enabling better manipulation and insight into the problem at hand.
Remember, always simplify before trying to solve an inequality or equation, as this reduces complexity and potential mistakes.