Question

Which statements are true about the solution 2 < x - 7 and its graph? Check all that apply.

x < 9
x > 9
The graph has a closed circle.
The graph has an open circle.
The arrow points left.
The arrow points right.

Answer 1

To analyze the solution 2 < x - 7 and its graph, let's break it down:

2 < x - 7

First, let's solve for x:

Add 7 to both sides of the inequality:

2 + 7 < x

9 < x

So, the solution to the inequality is x > 9.

Now, let's analyze the statements:

x < 9: This statement is not true because the solution to the inequality is x > 9, not x < 9.

x > 9: This statement is true since the solution to the inequality is x > 9.

The graph has a closed circle: The graph does not have a closed circle because the inequality is strict (2 < x - 7), indicating that the endpoint is not included in the solution. Therefore, the graph has an open circle.

The graph has an open circle: This statement is true since the graph has an open circle due to the strict inequality.

The arrow points left: The arrow does not point left because the solution is x > 9, indicating that x is greater than 9, which is to the right of the number line.

The arrow points right: This statement is true since the arrow points right, indicating that the values of x are greater than 9.

In summary, the true statements about the solution 2 < x - 7 and its graph are:

x > 9

The graph has an open circle

The arrow points right

Answer 2

Before we decide which statements are true, we will solve the inequality,

2 < x - 7.

If I transpose 7 to the left, with the plus sign, I get:

2 + 7 < x

9 < x

To make it more convenient, I rewrite it as x > 9.

Now we can deduce which statements are true and which ones aren't.

We already know that x > 9 is true.

Now let's talk about open and closed circles.

When graphing inequalities, we use an arrow, and a circle; the latter can be open or closed.

It is open, if we have less / greater than. Such is the case right here.

Closed circles symbolise inclusion, i.e., the solution is included into the solution set (SS). To show that a solution is included, we use the signs < (aka less than or equal to), or >, aka greater than or equal to.

Based on these pieces of information, let's consider this:

With x > 9, will the circle be open or closed?

The sign is all we need, because it provides all the info needed for answering the question.

The sign is greater than, not equal to; henceforth the circle is an open one.

And then last but not least, the arrow. It is important to note its direction because if it points left, the solutions are less than a specific point (the arrow goes from that specific point into infinity). The "specific point" is the solution to the inequality.

Since it's "greater than" the arrow points right;

`\stackrel{9}\bigcirc\!\!\!\!-\!\!\!-\!\!-\!\!-\!\!\!\-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!-\!\!\!\!\! >`

This is a diagram.