Problem 8 - Elementary Algebra

Question

Determine whether the given number is a solution to the given inequality. $$ x<0 \text { or } x \geq 1 ; x=12 $$

Step-by-Step Solution

Answer

Yes, 12 is a solution since it satisfies \( x \geq 1 \).

Step 1: Understanding the Inequality

We are given the inequality \( x < 0 \) or \( x \geq 1 \). This means that \( x \) must either be less than 0 or greater than or equal to 1 to satisfy the inequality.

Step 2: Identifying the Test Value

The number we are testing is \( x = 12 \). We will check if this value satisfies either part of the inequality \( x < 0 \) or \( x \geq 1 \).

Step 3: Checking the First Condition

First, we check \( x < 0 \). Substituting \( x = 12 \), we have \( 12 < 0 \), which is false. So, the number does not satisfy the first condition.

Step 4: Checking the Second Condition

Now, we check \( x \geq 1 \). Substituting \( x = 12 \), we have \( 12 \geq 1 \), which is true. Therefore, the number satisfies the second condition.

Step 5: Conclusion

Since the test value \( x = 12 \) satisfies one part of the inequality, namely \( x \geq 1 \), the number is indeed a solution to the inequality \( x < 0 \) or \( x \geq 1 \).

Key Concepts

Understanding Inequality Conditions

Inequalities are like little puzzles where we find numbers that make a statement true. In this exercise, we are given the inequality \( x < 0 \) or \( x \geq 1 \). This inequality is made up of two parts, and this is key to understanding it.What does it mean?

• \( x < 0 \): This tells us that \( x \), the number we are looking for, must be less than 0. It should be a negative number or exactly zero.
• \( x \geq 1 \): This tells us that \( x \) must be greater than or equal to 1. So, \( x \) could be 1, or 2, or any number larger than 1.

These conditions show that \( x \) has two different paths to follow in order to be a solution. It must satisfy one of them for it to be a correct fit. Think of it as a fork in the road where \( x \) needs to pick a lane.

Using the Substitution Method

The substitution method is a straightforward way to check if a certain number solves an inequality. Let's see how we can use it in this context!Checking the number 12:

• We substitute \( x = 12 \) into the inequality.
• First, we try \( x < 0 \): Substituting gives us \( 12 < 0 \). But, 12 is not less than 0 — this part doesn't work!
• Next, we check \( x \geq 1 \): Now, substituting gives \( 12 \geq 1 \). This is true because 12 is indeed greater than 1.

The substitution method helps us see, step by step, how each part of the inequality compares with our test number. It's like trying on clothes to see if they fit!

Conducting Solution Verification

After using the substitution method, it's essential to verify the solution thoroughly. Verification ensures that we understand not just whether the number works, but why it works, which is great for deeper learning.Was 12 a solution?

• Yes, 12 satisfies the second part of the inequality \( x \geq 1 \). Therefore, 12 is a solution.
• If the number does not satisfy any part, it isn't a solution. However, in this situation, satisfying just one part is enough due to the 'or' condition.
• Ensure every step is checked with logic for maximum accuracy.

By verifying the solution, we're reinforcing how inequalities function and affirming our understanding. This way, next time you're looking at inequalities, the process will be even clearer! Feel confident as you verify, knowing you've covered all bases.