Problem 13 - Prealgebra

Question

Solve each inequality. Check your solution. $$144<12 d$$

Step-by-Step Solution

Answer

The solution is \(d > 12\).

Step 1: Isolate the Variable

Start by isolating the variable in the inequality. The original inequality is \[ 144 < 12d. \] To isolate \(d\), divide both sides of the inequality by 12. \[\frac{144}{12} < \frac{12d}{12}. \] Simplifying gives: \[ 12 < d.\] This tells us that \(d\) must be greater than 12.

Step 2: Check the Solution

To verify the solution, choose a value for \(d\) that is greater than 12. Let's use \(d = 13\). Substitute \(d = 13\) back into the original inequality:\[ 144 < 12 imes 13. \]Calculate the right side of the inequality:\[ 12 imes 13 = 156. \]The inequality becomes\[ 144 < 156, \]which is a true statement. This confirms that our solution is correct.

Key Concepts

Isolating the Variable

Solving inequalities starts with the crucial step of isolating the variable. This aids in understanding which values the variable can take. In our exercise, the inequality given was \(144 < 12d\). With such an equation, the goal is to have the variable \(d\) alone on one side of the inequality. This can be achieved by following these basic steps:

• Identify what needs to be done to isolate the variable. Here, \(d\) is multiplied by 12.
• Use the inverse operation to cancel out this multiplication, which is division in this case.

By dividing both sides of the inequality by 12, you effectively "undo" the multiplication. So, \(\frac{144}{12} < \frac{12d}{12}\), leading to \(12 < d\).

This shows that for the inequality to remain true, \(d\) must be greater than 12. Isolating the variable simplifies the inequality, making it much easier to understand what values \(d\) can take.

Verifying the Solution

Once the variable is isolated, it's important to verify that the solution found is correct. Verification ensures that we haven't made any mistakes during calculations and guarantees the integrity of the solution. For the inequality \(12 < d\), any number greater than 12 should satisfy the inequality. Here's how to verify it:

• Select a convenient value for \(d\) that is greater than 12, such as \(d = 13\).
• Substitute this value back into the original inequality \(144 < 12d\).

When we substitute \(d = 13\), we calculate \(12 \times 13 = 156\), then check if \(144 < 156\).

Since this statement is true, our inequality solution is verified. Ensuring verification gives you confidence that \(d\) values like 13 indeed satisfy the original inequality settings.

Solve Step by Step

Approaching inequalities step by step makes the solving process straightforward and manageable. Each step should build upon the previous one, creating a clear and logical path to the solution. Here's a refined approach for our inequality \(144 < 12 d\):

• Isolate the variable: Divide both sides by 12 to work towards \(d\) alone, getting \(12 < d\).
• Choose a test value: Select any number greater than 12 to verify, like 13.
• Substitute and evaluate: Replace \(d\) with your test value in the original inequality to see if it holds true.

Breaking down the process ensures no step is overlooked, and each part is understood before proceeding. Solving systematically aids in internalizing the method, making it easier to tackle similar problems in the future.