Let's solve the given system of inequalities step-by-step:
1. Inequalities provided are:
[tex]\[
y \leq 2x - 42
\][/tex]
[tex]\[
y - x \geq 2
\][/tex]
2. Rewrite the second inequality:
To interpret the second inequality in the form similar to the first, we can rearrange it:
[tex]\[
y - x \geq 2
\][/tex]
Adding [tex]\(x\)[/tex] to both sides of the inequality to isolate [tex]\(y\)[/tex] on the left:
[tex]\[
y \geq x + 2
\][/tex]
3. Interpret the system of inequalities:
Now we have the system of inequalities as:
[tex]\[
y \leq 2x - 42
\][/tex]
[tex]\[
y \geq x + 2
\][/tex]
4. Graphical representation:
To better understand the solution, imagine plotting these lines on a coordinate plane.
- The first inequality [tex]\( y \leq 2x - 42 \)[/tex] represents all the points on or below the line [tex]\( y = 2x - 42 \)[/tex].
- The second inequality [tex]\( y \geq x + 2 \)[/tex] represents all the points on or above the line [tex]\( y = x + 2 \)[/tex].
5. Determine the region of interest:
The region that satisfies both inequalities is the area where the lines [tex]\( y = 2x - 42 \)[/tex] and [tex]\( y = x + 2 \)[/tex] overlap.
6. Conclusion:
Therefore, the solution to the system of inequalities is the region where [tex]\(y\)[/tex] lies between the lines [tex]\( y = 2x - 42 \)[/tex] and [tex]\( y = x + 2 \)[/tex].
In other words, [tex]\(y\)[/tex] must be greater than or equal to [tex]\( x + 2 \)[/tex] and less than or equal to [tex]\( 2x - 42 \)[/tex]. This can be summarized as:
[tex]\[
y \text{ lies between the lines } y = 2x - 42 \text{ (upper boundary) and } y = x + 2 \text{ (lower boundary)}.
\][/tex]
