Problem 13 - Algebra 2

Question

Graph each absolute value inequality. $$ y \leq|4-x| $$

Step-by-Step Solution

Answer

Plot the V-shaped graph of the line \(y = 4 - x\) for \(x \leq 4\) and \(y = x - 4\) for \(x > 4\). The solution to the inequality is the region below this V-shaped graph, including the boundary line.

Step 1: Understand the Absolute Value Inequality

Begin by recognizing that an absolute value inequality like \(y \leq |4 - x|\) describes all the points \((x, y)\) on the coordinate plane such that the y-coordinate is less than or equal to the absolute value of \(4 - x\). The boundary of this inequality will be the graph of the equation \(y = |4 - x|\).

Step 2: Graph the Boundary Equation

To graph the equation \(y = |4 - x|\), consider two cases. Case 1: When \(x \leq 4\), the absolute value simplifies to \(y = 4 - x\). Case 2: When \(x > 4\), the absolute value simplifies to \(y = x - 4\). Plot the two pieces of the V-shaped graph on the coordinate plane.

Step 3: Determine the Solution Region

Since the inequality is \(y \leq |4 - x|\), the region that satisfies this condition is below the boundary equation that was graphed in Step 2. Shade the entire region below the V-shaped graph.

Key Concepts

Understanding Absolute Value Inequalities

Graphing an absolute value inequality involves visualizing a set of all points that satisfy a given condition related to distance away from a certain value. The absolute value symbol, denoted by vertical bars as in \( |x| \), represents the distance of a number \(x\) from zero on the number line, irrespective of direction. An absolute value inequality, such as \(y \leq |4 - x|\), dictates that \(y\) is less than or equal to the distance of \(4 - x\) from zero.

This means we're looking at all the \(y\)-values that are on or below the distance from \(4\) represented on the \(x\)-axis. Importantly, because distance can't be negative, the inequality splits into two scenarios that are addressed with piecewise functions. The boundary equation of the inequality \(y = |4 - x|\) forms a 'V' shape when graphed, and it's key to identify this shape to correctly graph the inequality region.

Navigating the Coordinate Plane

The coordinate plane is a two-dimensional surface where each point is defined by a pair of numbers \((x, y)\). These numbers are known as the coordinates, representing the horizontal (x-axis) and vertical (y-axis) displacements from the origin, which is the center of the plane where the axes intersect.

When it comes to graphing inequalities like \(y \leq |4 - x|\), the coordinate plane allows us to represent solution sets visually. By plotting the piecewise functions that define our boundary, we create a 'map' distinguishing regions where the inequality conditions are met. In the case of this inequality, the region of the plane that represents the solution is the area where any point \((x, y)\) would have a \(y\)-value that is on or below the V-shaped boundary line.

Piecewise Functions as Boundaries

Piecewise functions are essential for graphing absolute value inequalities because they allow us to define different expressions depending on the range of the input. In our exercise with \(y \leq |4 - x|\), we deal with two distinct linear expressions: \(4 - x\) when \(x \leq 4\), and \(x - 4\) when \(x > 4\). These linear equations are the branches of our V-shaped graph that form the boundary of our inequality.

Each branch is graphed separately, corresponding to a particular domain (value of \(x\)). After plotting these lines, the region that is a solution to the inequality is the space where the function lies below the V-shape, indicated by the \(y\)-values that are less than or equal to the absolute value expression. This portrays the piecewise nature of the function graphically, with each 'piece' applying to a different interval of the x-axis.