Question

Solve \(7 x-6 \leq 2 x+4\). A. \(x \leq 2\) B. \(x \geq 2\) C. \(x \leq 5\) D. \(x \leq 15\)

Short answer

A. \( x \leq 2 \)

Step-by-step solution

Move Variable Terms to One Side

Start by isolating the variable terms on one side of the inequality. Subtract \(2x\) from both sides: \[ 7x - 6 - 2x \leq 2x + 4 - 2x \] This simplifies to: \[ 5x - 6 \leq 4 \]

Isolate the Variable

Next, isolate the variable term by adding 6 to both sides: \[ 5x - 6 + 6 \leq 4 + 6 \] This simplifies to: \[ 5x \leq 10 \]

Solve for the Variable

Finally, solve for \(x\) by dividing both sides by 5: \[ \frac{5x}{5} \leq \frac{10}{5} \] This simplifies to: \[ x \leq 2 \]

Key concepts

Solving Inequalities

Inequalities help us to determine the range of possible values that a variable can take. When solving inequalities, our goal is to isolate the variable on one side of the inequality symbol. This process is quite similar to solving equations, with only a few additional rules to consider. The key steps involve moving all variable terms to one side and constant terms to the other. It is also important to remember that multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol. For instance, solving the inequality from the problem: 1. Start by moving the variable terms to one side: 7x - 6 \leq 2x + 4 Subtract 2x from both sides: \[7x - 6 - 2x \leq 2x + 4 - 2x\] This simplifies to: \[5x - 6 \leq 4\] 2. Next, isolate the variable term by adding 6 to both sides: \[5x - 6 + 6 \leq 4 + 6\] This simplifies to: \[5x \leq 10\] 3. Finally, solve for \(x\) by dividing both sides by 5: \[\frac{5x}{5} \leq \frac{10}{5}\] This simplifies to: \[x \leq 2\] By following these steps meticulously, you can unravel most inequality problems.

Algebraic Manipulation

Algebraic manipulation forms the foundation of solving math problems, including inequalities. This involves techniques to simplify and rearrange equations and inequalities to find the solution. For inequalities, you'll often:

• Combine like terms to simplify expressions
• Use addition or subtraction to isolate the variable
• Multiply or divide to solve for the variable

Consider the example we solved:

• First, we moved all \(x\) terms to one side: \(7x - 6 \leq 2x + 4\)
• We subtracted \(2x\) from both sides: \(5x - 6 \leq 4\)
• We then isolated the \(x\) term by adding 6: \(5x \leq 10\)
• Finally, we divided both sides by 5 to find \(x \leq 2\)

Mastering algebraic manipulation ensures you have the tools to simplify and solve complex inequalities effectively. It’s like untangling a knot step by step until you are left with a clear solution.

GED Test Prep

Preparing for the GED test involves staying sharp with concepts like solving inequalities. Here are a few steps to excel:

• Understand the Basics: Ensure you are familiar with the steps to move variables and constants.
• Practice Regularly: Solve various inequality problems from textbooks and online resources.
• Review Mistakes: Go through incorrect solutions to understand where you went wrong and how to correct it.
• Memorize Key Rules: Remember the key rule about reversing the inequality sign when multiplying or dividing by a negative number.

By following these steps, you’ll build a strong foundation and increase your confidence to tackle inequalities questions on the GED test.”