Question

Determine whether the given point satisfies each inequality.

$3x+2y\le 10; \left(2,-1\right)$

Short answer

The point $\left(2,-1\right)$ satisfy the inequality $3x+2y\le 10$.

Step-by-step solution

Step-1 – Apply the concept of linear inequality and its graphing

A linear inequality is like a linear equation but with an inequality sign instead of equal to sign. For example, $x+y\le 3$ is a linear inequality and $x+y=3$ is the related linear equation.

A point that satisfies the inequality makes the inequality true otherwise inequality will be false.

The steps to graph the inequality are provided below.

1. If the inequality contains greater than or less than sign then the boundary of the line will be dashed. If the inequality contains signs of greater than or equal to or less than or equal to then the boundary of the line will be solid.

2. Select a point (known as test point) from the plane that does not lie on the boundary on the line and substitute it in the inequality.

3. If the inequality is true then shade the region that contains the test point otherwise shade the other region when inequality is false.

Step-2 – Interpret the sign of the inequality

Consider the inequality provided below.

$3x+2y\le 10$

The inequality contains the sign of greater than or equal to.

Therefore, the boundary line will be solid.

Step-3 – Checking the point 

The corresponding equation is $3x+2y\le 10$ and the given point is $\left(2,-1\right)$.

Substitute $x=2$and $y=-1$and check whether the inequality is true or false.

$\begin{array}{c}3x+2y\le 10\\ 3\left(2\right)+2\left(-1\right)\le 10\\ 6-2\le 10\\ 8\le 10\end{array}$

This is true.

Thus, the given point satisfies the given inequality.

Graphical representation

The shaded region represents all the points that will satisfy the inequality and the given point is contained within the shaded region.

Hence, the point $\left(2,-1\right)$satisfy the inequality $3x+2y\le 10$.