Problem 57 - Algebra A Combined Function

Question

Solve each inequality. Write each answer using solution set notation. $$ -5 x+4 \leq-4(x-1) $$

Step-by-Step Solution

Answer

The solution set is \( \{ x \mid x \geq 0 \} \).

Step 1: Expand the Expression

First, let's expand the expression on the right-hand side of the inequality. To do this, distribute the \(-4\) across the term \((x-1)\):\(-4(x-1) = -4x + 4\).

Step 2: Rewrite the Inequality

Now, rewrite the original inequality by replacing the expanded expression from Step 1:\(-5x + 4 \leq -4x + 4\).

Step 3: Isolate the Variable

Next, we want to bring all the \(x\) terms to one side of the inequality. Start by adding \(4x\) to both sides of the inequality:\(-5x + 4 + 4x \leq -4x + 4 + 4x\).Simplify to get:\(-x + 4 \leq 4\).

Step 4: Solve for the Variable

To isolate \(x\), subtract \(4\) from both sides:\(-x + 4 - 4 \leq 4 - 4\).Simplify to get:\(-x \leq 0\).

Step 5: Handle the Negative Variable Coefficient

To remove the negative sign from \(x\), multiply both sides of the inequality by\(-1\). Remember, multiplying or dividing an inequality by a negative number reverses the inequality symbol:\(x \geq 0\).

Step 6: Write the Solution in Set Notation

Finally, express the solution in solution set notation, accounting for all possible values:\\[ \{ x \mid x \geq 0 \} \].

Key Concepts

solution set notation

Solution set notation is a way to describe the set of all possible solutions to an inequality. Instead of listing each solution individually, which can be impossible in some cases, we use set notation to express the range of values for which the inequality holds true. This notation uses curly braces \(\{\}\) to define the set and includes a vertical bar \(|\) to mean "such that." For the inequality solution \(x \geq 0\), the solution set is written as \(\{ x \mid x \geq 0 \}\). This reads as "the set of all \(x\) such that \(x \) is greater than or equal to 0." In this notation:

• \(x\): refers to the elements in the set.
• \(\mid\): indicates a condition, meaning "such that."
• \(x \geq 0\): describes the condition that \(x\) satisfies.

This concise method provides a clear visual representation of the solution, making it easier for learners to understand and communicate mathematical ideas. Always remember to check the type of inequality in question, as the symbols \(\geq\), \(\leq\), \(<\), and \(>\) play a crucial role in determining the set's definition.

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are the building blocks of algebra and are used to represent mathematical relationships and solve problems like inequalities. An algebraic expression can consist of:

• **Constants**: Numbers on their own, like 4 or -7.
• **Variables**: Symbols that stand for unknown numbers, typically \(x\), \(y\), etc.
• **Operations**: Addition, subtraction, multiplication, and division.

In our examples, expressions like \(-5x + 4\) and \(-4(x-1)\) use these components. They involve operations like distribution and combining like terms:

• Expansion: Opening brackets using the distributive property, as seen with \(-4(x-1)\), which becomes \(-4x + 4\).
• Combining Terms: Simplifying expressions by adding/subtracting like terms. Example: aligning and simplifying \(-5x + 4 + 4x\).

These techniques ease the complexity of problems, allowing us to manipulate expressions systematically in reaching a solution. Understanding how to build and simplify these expressions aids students in navigating through algebraic equations and inequalities effectively.

isolating variables

Isolating the variable is a critical step in solving inequalities or equations. The goal is to get the variable by itself on one side of the inequality, which often involves reversing operations that are applied to it. Here's a simple breakdown of how this works: 1. **Identify Terms Involving the Variable**: Look for all terms that include the variable, such as \(-5x\) in our inequality. 2. **Move Terms**: Use addition or subtraction to move all variable terms to one side and constant terms to the other side. In the problem, adding \(4x\) to both sides consolidates \(x\) terms on one side, leading to \(-x + 4 \leq 4\). 3. **Simplify & Solve**: To isolate \(x\), continue with algebra operations. For example, subtract 4 from each side, giving \(-x \leq 0\). 4. **Adjust for Negative Coefficients**: If your variable has a negative coefficient, like \(-x\), multiply the entire inequality by \(-1\). Remember, flipping the inequality sign is essential: \(x \geq 0\). By breaking down these steps, isolating variables becomes manageable. This notion is foundational in algebra, aiding students in successfully manipulating and solving their inequalities and equations. Understanding these principles paves the way for more complex problem-solving in new algebraic contexts.