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How to Graph the One-Variable Inequality $x \leq -2$ on a Number Line
Mathematics
Grade 8 (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 solid 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 equal $-2$ and all values less than $-2$.
2
Step 2: Choose the correct dot type. Since the inequality includes equality ($\leq$), we use a closed (filled) dot at $x=-2$ on the number line to show that $-2$ is part of the solution set.
3
Step 3: Draw the solution line. Because we are including all values less than $-2$, we draw a solid line extending to the left from the closed dot at $-2$, covering all numbers smaller than $-2$ on the number line.
Knowledge Points Involved
1
Graphing one-variable inequalities on a number line
This is a method to visually represent the solution set of an inequality with one variable. The number line acts as a scale, and we use dots and lines to mark all values that satisfy the inequality.
2
Closed vs. open dots for inequalities
A closed (filled) dot is used when the inequality includes equality ($\leq$ or $\geq$), indicating that the exact point on the number line is part of the solution. An open dot is used for strict inequalities ($<$ or $>$), meaning the exact point is not part of the solution set.
3
Direction of the solution line on a number line
For inequalities where the variable is less than a value ($x < a$ or $x \leq a$), the solution line extends to the left from the marked point. For inequalities where the variable is greater than a value ($x > a$ or $x \geq a$), the line extends to the right.
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