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Chapter 1: Problem 11

Graph the inequality: \(\mathrm{y} \geq \mathrm{x}-1\).

Short Answer

Expert verified
To graph the inequality \(y \geq x-1\), first graph the line \(y = x-1\) using its slope (1) and y-intercept (-1). Plot the points (0, -1) and (1, 0), and draw a solid line through them. Since the inequality is \(y \geq x-1\), shade the region above the line to represent all points where y is greater than or equal to x-1.

Step by step solution

01

Identify the equation, slope, and y-intercept

First, let's start by identifying the equation that represents the boundary of the inequality, and its slope and y-intercept. The equation is y = x-1, which is in the slope-intercept form (y = mx + b). From here, we can see that the slope (m) is 1, and the y-intercept (b) is -1.
02

Graph the equation y = x-1

To plot an equation on the Cartesian plane, we'll start by plotting the y-intercept (0, -1), which is the point on the y-axis where the line intersects. Next, we'll use the slope to find another point on the line. Since the slope is 1, this means that for each 1 unit we go to the right on the x-axis, we go up 1 unit on the y-axis. From the y-intercept, go one unit to the right and one unit up to find the second point (1,0). Plot these points on the Cartesian plane and draw a solid line that goes through both points. The solid line represents the equation y = x-1.
03

Determine the shading for the inequality

Now that we've plotted the equation y = x-1, we need to shade the region of the inequality. Since the inequality is y 鈮� x-1, we are looking for all points (x, y) where the y value is greater than or equal to x-1. To visualize this, imagine testing several points on either side of the line. For example, take the point (0,0), which is above the line: 0 鈮� 0 - 1 0 鈮� -1 (True) Since the inequality holds true for this point above the line, shade the region above the line to represent the points that satisfy the inequality.
04

Present the graph of the inequality

Now we have everything we need to present the final graph of the inequality. Plot the line y = x-1 on the Cartesian plane, and shade the region above the line. Label the graph with the inequality y 鈮� x-1.

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