Problem 42 - Algebra A Combined Function

Question

Solve each inequality. Write each answer using solution set notation. $$ 2 x-1 \geq 4 x-5 $$

Step-by-Step Solution

Answer

\(\{ x \,|\, x \leq 2 \}\)

Step 1: Move Variable Terms to One Side

We start by moving the variable terms to the same side of the inequality. Subtract \(2x\) from both sides of the inequality: \(2x - 1 - 2x \geq 4x - 5 - 2x\). This simplifies to: \(-1 \geq 2x - 5\).

Step 2: Isolate Constant Term

Next, we isolate the constant term by adding 5 to both sides of the inequality: \(-1 + 5 \geq 2x - 5 + 5\). This simplifies to: \(4 \geq 2x\).

Step 3: Solve for the Variable

Now, isolate the variable \(x\) by dividing both sides by 2: \(\frac{4}{2} \geq \frac{2x}{2}\). This results in: \(2 \geq x\).

Step 4: Write the Solution in Set Notation

Rewrite the inequality solution using solution set notation. Since \(x\) must be less than or equal to 2: \(\{ x \,|\, x \leq 2 \}\).

Key Concepts

Solution Set Notation

When we solve inequalities, we often express our answer using solution set notation. This notation provides a clear, concise way to describe all possible solutions.

In the given exercise, the solution set is expressed as \( \{ x \mid x \leq 2 \} \). Here's a breakdown of this notation:

• The curly braces \( \{ \} \) indicate that we're describing a set of values.
• The vertical bar or 'pipe' symbol \( \mid \) means 'such that'.
• The expression inside, \( x \leq 2 \), shows the condition that must be true for any number \( x \) included in the set. This inequality tells us that all values of \( x \) must be less than or equal to 2.

Solution set notation is very helpful in mathematics because it allows us to express an infinite number of solutions in a clear and organized way.

Algebraic Manipulation

Algebraic manipulation is a critical skill needed to solve inequalities and equations. It's the process of performing operations to move variables and constants around so that you can isolate the variable of interest.

In this exercise, the steps of algebraic manipulation include:

• **Moving terms**: We moved the \(2x\) term to the right side to consolidate like terms on one side of the inequality.
• **Isolating constants**: By adding 5 to both sides, we worked to isolate the constant terms to help solve for \(x\).
• **Solving for the variable**: Finally, by dividing both sides by 2, we isolated \(x\).

These steps are methodical and use foundational algebraic rules. They help simplify the inequality so that we can easily express our solution. Remember, maintaining the balance of the equation or inequality by doing the same operation on both sides is key.

Inequalities in Algebra

Understanding inequalities is essential in algebra because they describe a range of possible solutions rather than a single solution. Inequalities show how numbers relate to each other by using symbols such as \(>\), \(<\), \(\geq\), \(\leq\).

For the inequality \(2x - 1 \geq 4x - 5\), after simplifying and manipulating through algebraic steps, the final solution is \(x \leq 2\). This is different from an equality (like \(x = 2\)), where only one value satisfies the equation.

When solving inequalities:

• The direction of the inequality matters. For instance, if you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol.
• The solution often represents a set of numbers, and you usually use set notation to express this.

Inequalities are everywhere in math and everyday life. They are used to represent and solve problems where there is more than one possible solution, providing flexibility and range in exploration.