Exercise 7 Page 403

To determine if a point satisfies an equation, we substitute the point into the equation and simplify. If the resulting statement is true, then the point is contained in the solution set. For systems of inequalities we can use the same method. However, substituting the point must create true statements in every inequality in the system. Let's test the given point.

{y > 2 x + 4 y < 3 x + 7 (I) (II)

SubstituteII

$x = 2$ , $y = 12$

⎩ ⎪ ⎨ ⎪ ⎧ 12 > ? 2 (2) + 4 12 < ? 3 (2) + 7

$(I), (II):$ Multiply

⎩ ⎪ ⎨ ⎪ ⎧ 12 > ? 4 + 4 12 < ? 6 + 7

$(I), (II):$ Add terms

{12 > 8 ✓ 12 < 13 ✓

Since

12

is greater than

8

and less than

13,

both statements are true. Therefore, the point

(2, 12)

is contained in the solution set of the system.

Alternative Solution

Solve by Graphing

We could also verify that the point satisfies the system by graphing the system. When graphing a system of inequalities, we begin by drawing the boundary lines of the regions. In our case, the boundary lines are going to be dashed lines because both inequalities are strict.

In Equation (I), the $y -$ values must all be greater than $2 x + 4,$ so we need to shade above this line. In Equation (II), the $y -$ values must all be less than $3 x + 7,$ so we will to shade below this line.

Finally, we plot the given point and check whether or not it falls within the overlapping region.

It does! Therefore, we know that the point satisfies the given system.

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Alternative Solution