Problem 61 - Algebra 2

Question

Solve each equation or inequality. Check your solutions. $$ 2 x-4>8 $$

Step-by-Step Solution

Answer

The solution is \(x > 6\).

Step 1: Isolate the variable term

The given inequality is \(2x - 4 > 8\). To isolate the term with \(x\), add 4 to both sides of the inequality to remove the constant from the left side. This gives us \(2x > 12\).

Step 2: Solve for the variable

Now that we have \(2x > 12\), divide both sides by 2 to solve for \(x\). This yields \(x > 6\).

Step 3: Check the solution

Substitute a value greater than 6 back into the original inequality to verify the solution. For example, use \(x = 7\). Substitute into the original inequality: \(2(7) - 4 = 14 - 4 = 10\). Since \(10 > 8\), our solution \(x > 6\) is correct.

Key Concepts

Isolate Variable

To solve inequalities like the one we have, the first step is often to isolate the variable. This means we want the variable, in this case, \( x \), to be on one side of the inequality sign, all by itself. Start by looking at the original inequality: \( 2x - 4 > 8 \). You notice there’s a \(-4\) with the \( x \) term, which we need to move to the other side. You can do this by adding \( 4 \) to both sides. This operation cancels out the \(-4\) on the left side and modifies the right side from \( 8 \) to \( 12 \), giving us \( 2x > 12 \). Remember, whatever you do to one side of the inequality, you must do to the other, just like in an equation.

Isolating the variable is vital because it sets you up for solving the inequality completely. It simplifies the inequality to a form where you can directly solve for \( x \) in the next steps.

Check Solutions

Checking your solutions in math is like proof-reading a paper. It ensures you've actually got the right answer. Once you determine that \( x > 6 \), you'll need to test this to be sure. To check your solution, pick any number greater than \( 6 \) and plug it back into the original inequality, \( 2x - 4 > 8 \).

For instance, if we choose \( x = 7 \), substitute it back: \( 2(7) - 4 = 14 - 4 = 10 \). Since \( 10 > 8 \) holds true, the solution \( x > 6 \) is correct. This quick verification step not only confirms your solution but also boosts your confidence in your work. Always remember to check your solutions to avoid any careless mistakes!

Algebraic Manipulation

Algebraic manipulation is the process of rearranging equations or inequalities to make them easier to solve. In our problem, after isolating the variable term, we reached \( 2x > 12 \). The next logical step is to solve for \( x \) by dividing both sides by the coefficient of \( x \), which is \( 2 \). This gives us \( x > 6 \).

Here are some key techniques of algebraic manipulation to remember:

• Add or subtract terms to both sides to move constants around.
• Multiply or divide both sides by numbers to get the variable alone.
• Always perform the same operation on both sides of the equation or inequality to maintain balance.

These techniques are not only specific to solving inequalities but are also fundamental tools in all areas of algebra. Being proficient with algebraic manipulation allows you to solve more complex problems over time.