Let's break this down step by step and ensure everything is clear.

1. Writing the inequality:

Neil starts with $20 and deposits $60 every week. If his account balance is at least $380, we can write the inequality as: $20 + 60x \geq 380$ where $x$ represents the number of weeks.

2. Solving the equation to find the boundary:

Now, we solve the equation $20 + 60x = 380$ to find the boundary value: $20 + 60x = 380$ Subtract 20 from both sides: $60x = 360$ Divide both sides by 60: $x = 6$

So, $x = 6$ is the boundary.

3. Is $x = 6$ included in the solutions to the inequality?

Since the inequality is $20 + 60x \geq 380$, this means that $x = 6$ is included because the inequality uses "greater than or equal to" (≥), which allows equality.

Thus, Yes, $x = 6$ is included in the solutions.

4. Testing $x = 5$ and $x = 7$:

Now, let's check if $x = 5$ and $x = 7$ make the inequality true or false.

When $x = 5$:

Substitute $x = 5$ into the inequality: $20 + 60(5) = 20 + 300 = 320$ Since $320$ is less than 380, the inequality is false for $x = 5$.

When $x = 6$ (boundary):

Substitute $x = 6$ into the inequality: $20 + 60(6) = 20 + 360 = 380$ Since $380 \geq 380$ is true, the inequality is true for $x = 6$.

When $x = 7$:

Substitute $x = 7$ into the inequality: $20 + 60(7) = 20 + 420 = 440$ Since $440 \geq 380$ is true, the inequality is true for $x = 7$.

Summary:

• $x = 6$ is included because it satisfies the inequality.
• For $x = 5$, the inequality is false.
• For $x = 7$, the inequality is true.

Would you like to go over more examples or clarify anything?

Additional Questions:

1. What would the equation be if Neil started with a different amount, say $30?
2. If Neil deposited $50 per week instead of $60, what would the equation look like?
3. How would the inequality change if Neil's account balance needed to be at least $400 instead of $380?
4. If the inequality were $20 + 60x > 380$, would $x = 6$ still be a solution?
5. How would you graph the inequality $20 + 60x \geq 380$ on a number line?

Tip: Always check if the inequality includes equality (≥ or ≤), as this can affect whether boundary values are included in the solution.