Graph the solution set of each system of inequalities or indicate that the system has no solution. {2x−5y≤103x−2y\u003e6\begin{cases} 2x - 5y \leq 10 \\ 3x - 2y \u003e 6 \end{cases} {2x−5y≤103x−2y\u003e6

Step 1: Identify each inequality and rewrite them in slope-intercept form (y = mx + b) to make graphing easier. For the first inequality, start with $$x - 4y \leq 8. $$Solve for $$y by $$isolating it on one side. Step 2: For $$x - 4y \leq 8$$, subtract $$x$$ from both sides to get $$-4y \leq -x + 8. $$Then divide every term by $$-4. $$Remember to reverse the inequality sign when dividing by a negative number, resulting in $$y \geq \frac{1}{4}x - 2. $$Step 3: For the second inequality, $$4x - 2y \u003e 12$$, isolate $$y by $$subtracting $$4x$$ from both sides: $$-2y \u003e -4x + 12. $$Then divide every term by $$-2$$, reversing the inequality sign to get $$y ). Then shade the region above the first line (since y$$ \geq$$) and below the second line (since y$$ \u003c$). Step 5: The solution set is the overlapping shaded region that satisfies both inequalities. If there is no overlap, then the system has no solution.

Ch. 5 - Systems of Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

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Blitzer 8th Edition

Ch. 5 - Systems of Equations and Inequalities

Problem 29

Chapter 6, Problem 29

Graph the solution set of each system of inequalities or indicate that the system has no solution.
$\begin{cases}\)2x - 5y $\leq$ 10 \\3x - 2y > 6\(\end{cases}$

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Step 1: Identify each inequality and rewrite them in slope-intercept form (y = mx + b) to make graphing easier. For the first inequality, start with $x - 4y \leq 8$. Solve for $y$ by isolating it on one side.

Step 2: For $x - 4y \leq 8$, subtract $x$ from both sides to get $-4y \leq -x + 8$. Then divide every term by $-4$. Remember to reverse the inequality sign when dividing by a negative number, resulting in $y \geq \frac{1}{4}x - 2$.

Step 3: For the second inequality, $4x - 2y > 12$, isolate $y$ by subtracting \$4x$ from both sides: $-2y > -4x + 12$. Then divide every term by $-2$, reversing the inequality sign to get $y < 2x - 6$.

Step 4: Graph the boundary lines $y = \frac{1}{4}x - 2$ (solid line because of $\leq$) and $y = 2x - 6$ (dashed line because of >). Then shade the region above the first line (since $y \geq$) and below the second line (since $y <$).

Step 5: The solution set is the overlapping shaded region that satisfies both inequalities. If there is no overlap, then the system has no solution.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Graphing Linear Inequalities

Graphing linear inequalities involves plotting the boundary line represented by the corresponding linear equation and then shading the region that satisfies the inequality. For '≤' or '≥', the boundary line is solid, indicating points on the line are included. For '<' or '>', the line is dashed, showing points on the line are excluded.

System of Inequalities

A system of inequalities consists of two or more inequalities considered simultaneously. The solution set is the intersection of the individual solution regions, representing all points that satisfy every inequality in the system. If no common region exists, the system has no solution.

Slope-Intercept Form and Boundary Lines

Converting inequalities to slope-intercept form (y = mx + b) helps in graphing by identifying the slope and y-intercept of the boundary line. This form makes it easier to draw the line accurately and determine which side to shade based on the inequality sign.

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Graph the solution set of each system of inequalities or indicate that the system has no solution. {2x−5y≤103x−2y>6\(\begin{cases}\)2x - 5y \(\leq\) 10 \\3x - 2y > 6\(\end{cases}\){2x−5y≤103x−2y>6

Verified video answer for a similar problem:

Key Concepts

Graphing Linear Inequalities

System of Inequalities

Slope-Intercept Form and Boundary Lines

Graph the solution set of each system of inequalities or indicate that the system has no solution.
$\begin{cases}\)2x - 5y \(\leq\) 10 \\3x - 2y > 6\(\end{cases}$