Problem 11 - Algebra 1: Concepts and Skills

Question

Determine whether the inequality is a multi-step inequality. Then explain how you would solve the inequality. $$ 2 w-1>6 w+2 $$

Step-by-Step Solution

Answer

Yes, it is a multi-step inequality. The solution to the inequality \(2w - 1 > 6w + 2\) is \(w < - \frac{3}{4}\).

Step 1: Identification of Multi-Step Inequality

The inequality \(2w - 1 > 6w + 2\) involves both subtraction and addition operations. Therefore, it is a multi-step inequality.

Step 2: Isolate variable w on one side of the inequality

In order to solve the inequality, first, try to get all the terms with 'w' on one side and constants on another side. For that, subtract '2w' from both sides of the inequality which results in \(-1 > 4w + 2\).

Step 3: Continue Isolation of variable

Next, subtract 2 from both sides, to remove '+2' from the right side. This results in \(-3 > 4w\).

Step 4: Complete Isolation of the variable

The last step to fully isolate 'w' is to divide every term by '4'. This provides the final solution, \(w < - \frac{3}{4}\)

Key Concepts

Solving Inequalities

When you come across an inequality like \(2w - 1 > 6w + 2\), solving it follows similar principles to solving equations. However, inequalities introduce the concept of "greater than" or "less than," adding an extra layer to the process. With multi-step inequalities, you have to move beyond simple operations.
Here are the general steps:

• Simplify both sides: Before isolating the variable, simplify both sides as much as possible. For our inequality, terms like \(2w\) and \(6w\) are on different sides, so consider moving variables to one side first.
• Isolate the variable: This involves rearranging the inequality until you have the variable on one side alone, using operations like addition, subtraction, multiplication, or division.
• Respect the inequality sign: When multiplying or dividing both sides by a negative number, flip the inequality sign. This is crucial when working with inequalities.

Understanding these steps will help ensure you tackle multi-step inequalities effectively.

Variable Isolation

Variable isolation is a strategic method in solving for an unknown within an equation or inequality. Imagine you are peeling away layers to find the core - the unknown variable. Let's break it down:

• Identify variables and constants: Distinguish between terms that contain the variable you are solving for and those that do not.
• Use inverse operations: To isolate a variable, reverse the operations associated with it. This means using addition to counteract subtraction, or division to counteract multiplication.
• Step-by-step isolation: In our example, after moving the \(2w\) term to one side, we subtracted 2 from both sides. Each step gradually simplifies the inequality until the variable is isolated.
• Verification: Double-check your work by substituting your solution back into the original inequality to verify the solution is legitimate.

Mastering variable isolation means knowing what actions to take at each step to solve for the variable in question.

Algebraic Expressions

Algebraic expressions consist of numbers, variables, and operations. They operate as building blocks in algebra and are essential in understanding inequalities.

The inequality \(2w - 1 > 6w + 2\) itself is comprised of two algebraic expressions, each with its variables and constants. Knowing how to simplify these expressions is crucial:

• Simplification: Simplify each term in the expression separately. Key tactics include combining like terms and factoring where possible.
• Balance the expressions: As you rearrange or simplify, keep both sides of the expression balanced. This means if you subtract a number from one side, do the same on the other.
• Translate theory into action: Algebraic expressions provide the framework to transition from complex problems to understandable steps, paving the way for solving effectively.

By understanding algebraic expressions, you can efficiently simplify, analyze, and solve inequalities and other algebraic problems with clarity.