In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x+y>4, x+y>−1

Verified step by step guidance

1. Step 1: Identify the system of inequalities given: $\{\begin{array}{c}{2}^x-y>4\\ 2^x-y>-5\end{array}$. Note that the inequalities are $2x\; -\; y\; >\; 4$ and $2x\; -\; y\; >\; -5$.
2. Step 2: Rewrite each inequality in slope-intercept form ($y\; =\; mx\; +\; b$) to better understand the boundary lines. For the first inequality, solve for $y$: $2x\; -\; y\; >\; 4$ becomes $-y\; >\; 4\; -\; 2x$, then multiply both sides by $-1$ (remember to reverse the inequality sign): . Similarly, for the second inequality, $2x\; -\; y\; >\; -5$ becomes .
3. Step 3: Graph the boundary lines $y\; =\; 2x\; -\; 4$ and $y\; =\; 2x\; +\; 5$. Since the inequalities are strict ($>$), use dashed lines to indicate that points on the lines are not included in the solution.
4. Step 4: Determine the solution region for each inequality. Because the inequalities are and , shade the region below each line. The solution to the system is the intersection of these shaded regions.
5. Step 5: Analyze the intersection of the two shaded regions. Since is more restrictive (lower) than , the solution set is the region below the line $y\; =\; 2x\; -\; 4$. This is the final graph of the solution set.

Key Concepts

Graphing Linear Inequalities

Graphing linear inequalities involves plotting the boundary line represented by the related linear equation and then shading the region that satisfies the inequality. The boundary line is dashed if the inequality is strict (>, <) and solid if it includes equality (≥, ≤). This visual representation helps identify all points that satisfy the inequality.

System of Inequalities

A system of inequalities consists of two or more inequalities considered simultaneously. The solution set is the intersection of the regions that satisfy each inequality individually. Graphing the system involves shading the overlapping region that meets all inequalities, or determining if no such region exists.

Interpreting Inequality Boundaries and Regions

Understanding how to interpret the boundary lines and the direction of shading is crucial. For inequalities like 2x - y > 4, rearranging to y < 2x - 4 helps identify which side of the line to shade. Correctly determining the feasible region ensures accurate graphing of the solution set.