Problem 40 - Prealgebra

Question

Solve each inequality. Check your solution. $$f-4 \geq 1.4$$

Step-by-Step Solution

Answer

The solution is \(f \geq 5.4\).

Step 1: Isolation of Variable

First, isolate the variable \(f\) in the inequality. To do this, add 4 to both sides of the inequality: \[f - 4 + 4 \geq 1.4 + 4\] This simplifies to \[f \geq 5.4\] So the solution is \(f \geq 5.4\).

Step 2: Verification of the Solution

To check the solution, substitute a value for \(f\) that satisfies the inequality. Let's try \(f = 5.5\) which is greater than 5.4. Substitute \(f = 5.5\) into the original inequality: \[5.5 - 4 \geq 1.4\] \[1.5 \geq 1.4\] This statement is true, confirming our solution is correct.

Key Concepts

Isolation of Variable

When you encounter an inequality like \( f - 4 \geq 1.4 \), your first step should be to isolate the variable, which in this case is \( f \). Isolating the variable means getting \( f \) by itself on one side of the inequality. This is similar to solving an equation but requires special attention to the inequality sign.

To isolate \( f \), add 4 to both sides of the inequality. This step counteracts the "-4" on the left side, which opposes the subtraction on \( f \). We perform this step without changing the direction of the inequality because we're only adding, not multiplying or dividing by a negative value.

Applying this, we have:

• \( f - 4 + 4 \geq 1.4 + 4 \)


• This simplifies to \( f \geq 5.4 \)

Now, \( f \) stands alone, and the inequality is much easier to understand!

Verification of Solution

Verification of the solution is crucial to ensure that the answer we found actually satisfies the original inequality.

After isolating the variable, we concluded that \( f \geq 5.4 \). To verify this, we need to choose a value for \( f \) that is greater than 5.4. Let's pick \( f = 5.5 \) as an example. We substitute \( f = 5.5 \) back into the original inequality to check if it holds true.

Substituting, we have:

• \( 5.5 - 4 \geq 1.4 \)
• This simplifies to \( 1.5 \geq 1.4 \), which is true.

With this verification, we confirm that our solution \( f \geq 5.4 \) is correct. If our chosen value did not satisfy the inequality, it would indicate an error in earlier steps.

Solving Inequalities

Solving inequalities involves finding all possible values for a variable that makes the inequality statement true. It's similar to solving equations but with a twist - we must keep an eye on the direction of the inequality sign.

Here are key points to remember:

• Addition or subtraction on both sides does not change the inequality's direction.


• Multiplying or dividing both sides by a positive number keeps the inequality direction unchanged.


• Multiplying or dividing both sides by a negative number reverses the inequality direction.

In the given example, where \( f - 4 \geq 1.4 \), only addition was involved, so the direction of the inequality remained the same throughout the process. Understanding these rules helps prevent common mistakes while solving inequalities.