Question

solve the system of inequalities
1)
$4x + y < 2$
$y > -2$

Explanation:

Step1: Analyze \( y > -2 \)

The inequality \( y > -2 \) represents the region above the horizontal line \( y = -2 \). This line is dashed (since the inequality is strict, \( > \), not \( \geq \)).

Step2: Analyze \( 4x + y < 2 \)

Rewrite \( 4x + y < 2 \) as \( y < -4x + 2 \). The boundary line is \( y = -4x + 2 \), which is dashed (because the inequality is strict, \( < \), not \( \leq \)). To graph this, find two points: when \( x = 0 \), \( y = 2 \); when \( y = 0 \), \( x=\frac{2}{4}=0.5 \). Draw the dashed line through \( (0, 2) \) and \( (0.5, 0) \), then shade the region below this line (since \( y < -4x + 2 \)).

Step3: Find the Intersection

The solution to the system is the region that is above \( y = -2 \) (dashed line, shade above) and below \( y = -4x + 2 \) (dashed line, shade below). This is the overlapping region of the two shaded areas.

To describe the solution set:

• For the line \( y = -2 \) (dashed), shade above it.
• For the line \( y = -4x + 2 \) (dashed), shade below it.
• The solution is the area where both shadings overlap.

In set - builder notation, the solution is \( \{(x,y)\mid y > - 2\text{ and }y < - 4x + 2\} \)

Answer:

The solution is the region above the dashed line \( y=-2 \) and below the dashed line \( y = - 4x + 2 \). In set - builder form, \( \boldsymbol{\{(x,y)\mid y > - 2\text{ and }y < - 4x + 2\}} \) (graphically, the overlapping region of the area above \( y=-2 \) and below \( y=-4x + 2 \) with both boundary lines dashed).