Question

Graph the solution set of each system of inequalities.﻿ex−y≤$$1e^{x}-y$$\le1ex−y≤1﻿﻿x−2y≥4x-2y\ge4x−2y≥4

Accepted Answer

Step 1: Start by graphing the first inequality, $$ e^x - y \leq 1 . To do $$this, first graph the boundary line $$ y = e^x - 1 . $$This is an exponential function shifted down by 1 unit. Plot several points for $$ x $$ values and connect them to form the curve. Step 2: Determine which side of the boundary line $$ y = e^x - 1  to $$shade. Choose a test point not on the line, such as (0,0). Substitute into the inequality: $$ e^0 - 0 \leq 1 $$ simplifies to $$ 1 \leq 1 $$, which is true. Therefore, shade the region below the curve. Step 3: Next, graph the second inequality, $$ x - 2y \geq 4 . $$Start by graphing the boundary line $$ x - 2y = 4 . $$Rearrange to slope-intercept form: $$ y = \frac{x}{2} - 2 . $$Plot the y-intercept at (0, -2) and use the slope $$ \frac{1}{2}  to $$find another point. Step 4: Determine which side of the line $$ y = \frac{x}{2} - 2  to $$shade. Use the test point (0,0) again: $$ 0 - 2(0) \geq 4 $$ simplifies to $$ 0 \geq 4 $$, which is false. Therefore, shade the region above the line. Step 5: The solution set of the system of inequalities is the region where the shaded areas from both inequalities overlap. Identify this region on the graph, ensuring to use dashed or solid lines appropriately based on whether the inequalities are strict or inclusive.

Graph the solution set of each system of inequalities.
$e^{x}-y\(\le\)1$
$x-2y\(\ge\)4$

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Key Concepts

Inequalities

Graphing Systems of Inequalities

Exponential Functions

Graph the solution set of each system of inequalities.ex−y≤1e^{x}-y\(\le\)1ex−y≤1x−2y≥4x-2y\(\ge\)4x−2y≥4