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How to Graph the One-Variable Inequality $x \leq -2$ 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 to place the correct dot on your line before you submit your answer.
Correct Answer
A closed (filled) dot at $x=-2$, with a line extending to the left (towards negative infinity) along the number line.
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 value 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 included in the solution set.
3
Step 3: Draw the solution line. Because $x$ can be any value less than $-2$, draw a solid line starting from the closed dot at $-2$ and extending to the left (towards the negative numbers on the number line).
Knowledge Points Involved
1
Graphing one-variable inequalities on a number line
This refers to representing the solution set of an inequality with a single variable using a number line. The graph uses dots and lines to show all values that satisfy the inequality. It is used to visually represent the range of valid solutions for linear inequalities in one variable.
2
Closed vs. open dots for inequalities
A closed (filled) dot is used when the inequality includes equality ($\leq$ or $\geq$), indicating that the endpoint value is part of the solution set. An open (hollow) dot is used for strict inequalities ($<$ or $>$), meaning the endpoint is not included in the solution set. This rule ensures accurate representation of the boundary of the solution range.
3
One-variable linear inequalities
These are inequalities that contain a single variable with a highest power of 1, such as $x \leq -2$. Their solution sets are ranges of real numbers, which can be easily visualized on a number line. They are foundational for understanding more complex inequalities and linear systems.
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