Q46. - Algebra 2

Question

Solve each system of inequalities by graphing.

46.  $\begin{array}{c} 2y+x\ge 4\\ y\ge x-4\end{array}$.

Step-by-Step Solution

Answer

The solution set of the given system of inequalities is shown in the below graph:

Step 1 – Sketch the solution of the inequality 2 y + x ≥ 4 .

Solve the equation $2y+x\ge 4$ for y:

  $\begin{array}{c}2y+x\ge 4\\ 2y\ge -x+4\\ y\ge -\frac{x}{2}+\frac{4}{2}\\ y\ge -\frac{x}{2}+2\end{array}$

The equation of the inequality $y\ge -\frac{x}{2}+2$ is $y=-\frac{x}{2}+2$ which is a line equation with slope $m=-\frac{1}{2},$ and the $y-$intercept is $\left(0,2\right)$.

Graphical interpretation of the given inequality $2y+x\ge 4$:

Step 2 – Sketch the solution of the inequality y ≥ x - 4 .

The equation of the inequality $y\ge x-4$ is $y=x-4$ which is a line equation with slope $m=1$ and the $y-$intercept is $\left(0,-4\right)$.

 

Graphical interpretation of the given inequality $y\ge x-4$:

Step 3 – Now sketch the solution set of the both the inequalities in the coordinate plane.

The solution set is the region colored in purple.