Question

related questions: lesson 5: inequalities

select the option that is true for the graph at 0.

q. \\(\frac{a}{10} \le 0\\) or \\(a - 6 > 0\\)

Explanation:

Solve the individual inequalities

We solve each part of the compound inequality separately.

First inequality:

$$ \frac{a}{10} \le 0 $$

Multiply both sides by \(10\):

$$ a \le 0 $$

Second inequality:

$$ a - 6 > 0 $$

Add \(6\) to both sides:

$$ a > 6 $$

Combine using the "or" logical operator

Using the Compound Inequalities knowledge point, the word "or" represents the union of the two solution sets.
The combined solution set is:

$$ a \le 0 \quad \text{or} \quad a > 6 $$

Analyze the graph at 0

Using the Graphing Compound Inequalities knowledge point, we determine the behavior of the graph at \(a = 0\).
The inequality \(a \le 0\) includes the value \(0\) because of the "less than or equal to" symbol (\(\le\)).
Therefore, at \(a = 0\), there should be a solid (closed) circle, and the line should be shaded to the left (towards negative infinity).

Determine the true statement

Since \(a = 0\) is a solution to the inequality \(a \le 0\), the statement "the graph has a closed circle at 0" or "the graph is shaded at 0" is true.
Specifically, for the boundary point \(0\):

• It uses a solid/closed circle because of the \(\le\) sign.
• The shading goes to the left of \(0\).

Answer:

The inequality simplifies to \(a \le 0\) or \(a > 6\).

For the graph at \(0\):

• There is a closed (solid) circle at \(0\).
• The number line is shaded to the left of \(0\).