Question

Solve each system of inequalities by graphing. 1

$\begin{array}{l}y-x>0\\ y+x>4\end{array}$

Short answer

The solution of the inequalities is$x<y<4-x.$

Step-by-step solution

Step-1 –Concept of solving the linear inequalities

To solve the inequalities we convert the inequalities into linear equations and find the solutions of the equations to obtain the graph.

Step-2 –Concept of shading the region

For shading the region, we choose a point. If the point satisfies the inequalities then the shaded region is towards the point otherwise, the shaded region is away from the point.

Step-3 –Solving the inequalities

Given inequalities are$y-x>0,y+x<4.$

Their respective linear equations are $y-x=0,y+x=4.$

The points which satisfy the equation$y-x=0,$are $\left(0,0\right)$and$\left(2,2\right)$

The points, which satisfy the equation $y+x=4$$\left(0,4\right)$and$\left(4,0\right)$

Step-4 –Evaluating the shaded region

We choose$\left(1,0\right)$to get the shaded region. The point$\left(1,0\right)$does not satisfy the inequality$y-x=0$but it satisfies the inequality$y+x=4$

Step-5 –Plotting the graph

Therefore, the graph for the inequalities is

The common shaded region is$x<y<4-x.$