Question

1. determine if the given points are solutions to the system of inequalities. justify your answer.

$y + x < 7$
$y - 2x \geq 1$

a. $(-2, -1)$
b. $(1, 6)$
c. $(3, 1)$

Explanation:

Step1: Analyze the system of inequalities

The system of inequalities is \( y + x < 7 \) and \( y - 2x \geq 1 \). We will check each point by substituting the \( x \) and \( y \) values into both inequalities.

Step2: Check point a: \((-2, -1)\)

Substitute into \( y + x < 7 \)

Substitute \( x=-2 \) and \( y = -1 \) into \( y + x \):
\( -1+(-2)=-3 \)
Since \( -3 < 7 \), this inequality is satisfied.

Substitute into \( y - 2x \geq 1 \)

Substitute \( x = -2 \) and \( y=-1 \) into \( y - 2x \):
\( -1-2\times(-2)=-1 + 4 = 3 \)
Since \( 3\geq1 \), this inequality is also satisfied. So \((-2, -1)\) is a solution.

Step3: Check point b: \((1, 6)\)

Substitute into \( y + x < 7 \)

Substitute \( x = 1 \) and \( y = 6 \) into \( y + x \):
\( 6 + 1=7 \)
But the inequality is \( y + x < 7 \), and \( 7\) is not less than \( 7 \), so this inequality is not satisfied. Thus, \((1, 6)\) is not a solution.

Step4: Check point c: \((3, 1)\)

Substitute into \( y + x < 7 \)

Substitute \( x = 3 \) and \( y = 1 \) into \( y + x \):
\( 1+3 = 4 \)
Since \( 4<7 \), this inequality is satisfied.

Substitute into \( y - 2x \geq 1 \)

Substitute \( x = 3 \) and \( y = 1 \) into \( y - 2x \):
\( 1-2\times3=1 - 6=-5 \)
Since \( -5<1 \), this inequality is not satisfied. So \((3, 1)\) is not a solution.

Answer:

• a. \((-2, -1)\) is a solution (satisfies both inequalities).
• b. \((1, 6)\) is not a solution (fails \( y + x < 7 \)).
• c. \((3, 1)\) is not a solution (fails \( y - 2x \geq 1 \)).