Question

solving systems of inequalities notes
solving systems of inequalities:

• make sure both inequalities are in ______________.
• graph the ____________ inequality and shade.
• graph the ______________ inequality and shade.
• find the ____________ in which they ____________.

inequality symbol	type of line (dashed or solid)	where to shade (above or below line)	are the points on the line part of the solution?
>
≤
≥

solve the system of inequalities

1. ( 4x + y < 2 )

( y > -2 )

graph with x-axis from -5 to 5 and y-axis from -5 to 5, grid lines, and axes labeled

Explanation:

Step1: Analyze \( 4x + y < 2 \)

Rewrite in slope - intercept form (\( y=mx + b \)): \( y<-4x + 2 \). The boundary line is \( y=-4x + 2 \) (dashed line, since \( < \)), shade below the line.

Step2: Analyze \( y > - 2 \)

The boundary line is \( y=-2 \) (dashed line, since \( > \)), shade above the line.

Step3: Find the intersection

The solution is the region where both shadings overlap. To graph:

• For \( y=-4x + 2 \), when \( x = 0 \), \( y = 2 \); when \( y = 0 \), \( x=\frac{1}{2} \). Draw dashed line through \((0,2)\) and \((\frac{1}{2},0)\), shade below.
• For \( y=-2 \), draw dashed horizontal line through \( y=-2 \), shade above.
• The overlapping region is below \( y=-4x + 2 \) and above \( y=-2 \).

Answer:

The solution region is the set of all points \((x,y)\) such that \( y < - 4x+2 \) and \( y > - 2 \), represented graphically as the area above \( y = - 2 \) (dashed line) and below \( y=-4x + 2 \) (dashed line).