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Magazine Chapter 2: Problem 6
Graph each inequality. $$ y \leq 3|x|-1 $$
Short Answer
Expert verified
Graph the V-shaped line and shade below it.
Step by step solution
01
Understand the Equation
The inequality given is \( y \leq 3|x| - 1 \). This means we are looking for the region where the y-values are less than or equal to \( 3|x| - 1 \). The equation \( y = 3|x| - 1 \) is V-shaped, symmetric about the y-axis due to the absolute value.
02
Graph the Boundary Line
First, graph the line \( y = 3|x| - 1 \). This involves plotting two lines: one for \( y = 3x - 1 \) for \( x \geq 0 \), and the other for \( y = -3x - 1 \) for \( x < 0 \). Draw this V-shaped graph on the coordinate plane, using a solid line because the inequality is \( \leq \). This indicates that points on the line are included in the solution.
03
Determine the Shaded Region
Since we have \( y \leq 3|x| - 1 \), the region we need is below the V-shaped line. Shade everything below or on the line \( y = 3|x| - 1 \) to represent \( y \leq 3|x| - 1 \).
04
Check a Solution Point
To ensure the shading is correct, choose a test point not on the line. For simplicity, use the origin \( (0,0) \). Substitute into the inequality: \( 0 \leq 3|0| - 1 \Rightarrow 0 \leq -1 \), which is false. This confirms the correct region is indeed below the line, as originally shaded.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Coordinate Plane
The coordinate plane is a two-dimensional surface where we can graphically represent mathematical expressions. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at the origin, a point represented by
- (0,0).
- (x, y),
- 'x' represents the horizontal distance from the origin
- and 'y' represents the vertical distance.
Absolute Value
Absolute value is a fundamental concept in mathematics. It refers to the distance of a number from zero on the number line, regardless of direction. The absolute value of a number
- 'a' is denoted by \( |a| \).
- \( |3| = 3 \)
- and \( |-3| = 3 \).
V-shaped Graph
The V-shaped graph is characteristic of equations involving absolute values. In the inequality \( y \leq 3|x| - 1 \), the graph appears as a V-shaped line. This occurs because:
- The equation splits into two parts due to the absolute value, one for positive x and one for negative x.
- It creates two separate linear equations: \( y = 3x - 1 \) for \( x \geq 0 \)
- and \( y = -3x - 1 \) for \( x < 0 \).
- These lines meet at the vertex of the V, which lies on the y-axis.
Shading Region
Shading is a visualization tool used in graphing inequalities. Once you have graphed the boundary line, such as \( y = 3|x| - 1 \), you need to identify which side of the line the solutions lie. In this case:
To verify correct shading:
- The inequality is \( y \leq 3|x| - 1 \), so the region you're interested in is below the boundary line.
To verify correct shading:
- Choose a test point, such as \( (0,0) \),
- and substitute it into the inequality.
- If the inequality holds, shade the side containing the point.
- If not, shade the opposite side.
