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How to Graph the Linear Inequality $y\\leq \\frac{1}{2}x - 1$
Mathematics
Grade 9 of Junior High School
Question Content
Graph the inequality: $y\\leq \\frac{1}{2}x - 1$
Correct Answer
A solid line representing $y=\\frac{1}{2}x - 1$ with the region below the line shaded
Detailed Solution Steps
1
Step 1: Identify the boundary line. The inequality is based on the linear equation $y=\\frac{1}{2}x - 1$. Since the inequality symbol is $\\leq$ (less than or equal to), the boundary line will be solid (to show that points on the line are included in the solution set).
2
Step 2: Graph the boundary line. Use the slope-intercept form $y=mx+b$: the y-intercept $b=-1$, so plot the point $(0, -1)$. The slope $m=\\frac{1}{2}$, meaning from the y-intercept, move up 1 unit and right 2 units to plot a second point $(2, 0)$. Draw a solid straight line through these two points.
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Step 3: Determine the region to shade. Choose a test point not on the line, such as $(0,0)$. Substitute into the inequality: $0\\leq \\frac{1}{2}(0)-1$ simplifies to $0\\leq -1$, which is false. Since the test point does not satisfy the inequality, shade the region on the opposite side of the boundary line (the region below the line).
Knowledge Points Involved
1
Linear Inequalities in Two Variables
These are inequalities that can be written in the form $Ax+By\\leq C$, $Ax+By\\geq C$, $Ax+By<C$, or $Ax+By>C$, where $A$, $B$, and $C$ are constants. Their solutions are represented as shaded regions on a coordinate plane, with a boundary line representing the corresponding linear equation.
2
Slope-Intercept Form of a Line
The form $y=mx+b$, where $m$ is the slope (rate of change between x and y values) and $b$ is the y-intercept (the point where the line crosses the y-axis). It is used to quickly graph linear equations by plotting the y-intercept and using the slope to find additional points.
3
Boundary Line Rules for Graphing Inequalities
For inequalities with $\\leq$ or $\\geq$, the boundary line is solid because points on the line are part of the solution set. For inequalities with $<$ or $>$, the boundary line is dashed because points on the line are not part of the solution set.
4
Test Points for Inequality Graphing
A point not on the boundary line that is substituted into the inequality to determine which region of the plane satisfies the inequality. If the substitution results in a true statement, the region containing the test point is shaded; if false, the opposite region is shaded.
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