Q 327 - Intermediate Algebra

Question

Graph the system $\left\{\begin{array}{l}x+2y\le 6\\ y\ge -\frac{1}{2}x-4\end{array}\right.$,

What does the solution mean?

Step-by-Step Solution

Answer

As one boundary line, the point of junction of the two lines is dashed.

The area coloured twice is the solution. This appears to be the deepest darkened area.

As a result, the purple zone is the answer.

Step 1. Given information

The given Linear inequalities are

$\left\{\begin{array}{l}x+2y\le 6\\ y\ge -\frac{1}{2}x-4\end{array}\right.$

Step 2. Graph of equation 1.

$$\begin{gathered} \mathrm{The} \mathrm{inequality} \mathrm{is} \mathrm{x}+2\mathrm{y}\le 6, \\ \mathrm{and} \mathrm{by} \mathrm{graphing} \mathrm{x}+2\mathrm{y}=6 \mathrm{and} \mathrm{testing} \mathrm{a} \mathrm{point}. \\ \mathrm{The} \mathrm{intercepts} \mathrm{ar}e x = 6 and \\ y = 3. \\ \mathrm{And} \mathrm{the} \mathrm{boundary} \mathrm{will} \mathrm{be} \mathrm{solid}. \\ \mathrm{Test} (0,0) \mathrm{which} \mathrm{makes} \mathrm{the} \mathrm{inequality} \mathrm{true}. \\ \mathrm{So} \mathrm{shade} \left(\mathrm{Blue}\right) \mathrm{the} \mathrm{side} \mathrm{that} \mathrm{does} \mathrm{contain} (0,0). \\ \mathrm{The} \mathrm{graph} \mathrm{for} \mathrm{the} \mathrm{equation} \mathrm{is}: \end{gathered}$$

Step 3. Graph for the equation 2

$$\begin{gathered} \mathrm{The} \mathrm{given} \mathrm{inequality} y\ge -\frac{1}{2}x-4 , \mathrm{and} \mathrm{by} \mathrm{graphing} \\ y\ge -\frac{1}{2}x-4 \mathrm{and} \mathrm{testing} \mathrm{a} \mathrm{point}. \\ A\mathrm{nd} \mathrm{the} \mathrm{boundary} \mathrm{will} \mathrm{be} \mathrm{solid}. \\ \mathrm{Test} (0,0) \mathrm{which} \mathrm{makes} \mathrm{the} \mathrm{inequality} \mathrm{true}. \\ \mathrm{So} \mathrm{shade} \left(\mathrm{blue}\right) \mathrm{the} \mathrm{side} \mathrm{that} \mathrm{does} \mathrm{contain} (0,0). \\ \mathrm{The} \mathrm{graph} \mathrm{for} \mathrm{the} \mathrm{linear} \mathrm{inequality} \mathrm{is} : \end{gathered}$$

Step 4. Common region in graph

The purple color is the area between the two linear inequalities.