Sure! Let's solve the system of inequalities step-by-step.
We are given the system of inequalities:
[tex]\[ y + 2x > 3 \][/tex]
[tex]\[ y \geq 3.5x - 5 \][/tex]
First, let's convert each inequality into its slope-intercept form, [tex]\( y = mx + b \)[/tex].
1. For the first inequality [tex]\( y + 2x > 3 \)[/tex]:
- We need to isolate [tex]\( y \)[/tex] on one side:
[tex]\[ y > -2x + 3 \][/tex]
- The first inequality, [tex]\( y + 2x > 3 \)[/tex], is equivalent to [tex]\( y > -2x + 3 \)[/tex].
- The inequality [tex]\( y > -2x + 3 \)[/tex] has a boundary line of [tex]\( y = -2x + 3 \)[/tex] in slope-intercept form.
2. For the second inequality [tex]\( y \geq 3.5x - 5 \)[/tex]:
- This inequality is already in slope-intercept form:
[tex]\[ y \geq 3.5x - 5 \][/tex]
- The inequality [tex]\( y \geq 3.5x - 5 \)[/tex] has a boundary line of [tex]\( y = 3.5x - 5 \)[/tex].
Now that we have rewritten the inequalities in slope-intercept form, let's analyze the regions defined by these inequalities:
- The first inequality [tex]\( y > -2x + 3 \)[/tex] defines a region above the line [tex]\( y = -2x + 3 \)[/tex].
- The second inequality [tex]\( y \geq 3.5x - 5 \)[/tex] defines a region above or on the line [tex]\( y = 3.5x - 5 \)[/tex].
The solution set to the system of inequalities is the intersection of the regions defined by [tex]\( y > -2x + 3 \)[/tex] and [tex]\( y \geq 3.5x - 5 \)[/tex]. This means we look for the area where both conditions are met simultaneously.
To verify if a point is in the solution set of the system of inequalities, it must satisfy both:
1. [tex]\( y > -2x + 3 \)[/tex]
2. [tex]\( y \geq 3.5x - 5 \)[/tex]
Hence, the given system of inequalities has a solution set that is shaded above both boundary lines. The inequalities in slope-intercept form are:
[tex]\[ y > -2x + 3 \][/tex]
[tex]\[ y \geq 3.5x - 5 \][/tex]
And the point that fulfills both inequalities will be in the solution set of the system.
