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How to Graph the Inequality x ≤ -2 on a Number Line
Mathematics
Grade 7 (Junior High School)
Question Content
Graph the inequality $x \leq -2$. Toggle the "Dot" or "Open Dot" button for your line before you submit your answer.
Correct Answer
A closed (filled) dot at $x=-2$ on the number line, with a line extending to the left (towards negative infinity) from this dot.
Detailed Solution Steps
1
Step 1: Analyze the inequality symbol. The inequality is $x \leq -2$, which means $x$ can be equal to $-2$ or any number less than $-2$.
2
Step 2: Choose the correct dot type. Since the inequality includes equality ($\leq$), we use a closed (filled) dot to mark the point $x=-2$ on the number line.
3
Step 3: Draw the line direction. Because $x$ can be values less than $-2$, we draw a solid line starting from the closed dot at $x=-2$ and extending to the left (towards smaller, more negative numbers) along the number line.
Knowledge Points Involved
1
Graphing linear inequalities on a number line
This involves representing the set of all numbers that satisfy a one-variable linear inequality. The graph uses dots and lines to show the valid range of values, which is foundational for understanding solution sets of inequalities.
2
Closed vs. open dots for inequalities
A closed (filled) dot is used when the inequality includes equality ($\leq$ or $\geq$), meaning the endpoint is part of the solution set. An open dot is used for strict inequalities ($<$ or $>$), meaning the endpoint is not part of the solution set.
3
Direction of the solution line on a number line
For inequalities like $x \leq a$ or $x < a$, the line extends to the left from the endpoint (towards smaller numbers). For $x \geq a$ or $x > a$, the line extends to the right (towards larger numbers).
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