Saif Qureshi

2022-06-27

Determine which point is part of the solution set to the following
system of inequalities: f(x) <x + 4; f(x) > -x - 3; and f(x) < 5.

Mr Solver

To determine which point is part of the solution set to the system of inequalities: $f\left(x\right), $f\left(x\right)>-x-3$, and $f\left(x\right)<5$, we need to evaluate each point and check if it satisfies all three inequalities.
Let's consider a point $\left(x,y\right)$ and substitute its coordinates into each inequality to see if it holds true.
1. For $f\left(x\right), we substitute $y$ into $f\left(x\right)$:
$y
2. For $f\left(x\right)>-x-3$, we substitute $y$ into $f\left(x\right)$:
$y>-x-3$
3. For $f\left(x\right)<5$, we substitute $y$ into $f\left(x\right)$:
$y<5$
Now, let's examine each point and see which one satisfies all three inequalities:
1. Point A: (2, 1)
Substituting the coordinates into the inequalities:
$1<2+4\phantom{\rule{1em}{0ex}}✓$
$1>-2-3\phantom{\rule{1em}{0ex}}✓$
$1<5\phantom{\rule{1em}{0ex}}✓$
Point A satisfies all three inequalities.
2. Point B: (-3, 6)
Substituting the coordinates into the inequalities:

$6>3-3\phantom{\rule{1em}{0ex}}✓$

Point B does not satisfy the first and third inequalities.
3. Point C: (0, -1)
Substituting the coordinates into the inequalities:
$-1<0+4\phantom{\rule{1em}{0ex}}✓$
$-1>0-3\phantom{\rule{1em}{0ex}}✓$
$-1<5\phantom{\rule{1em}{0ex}}✓$
Point C satisfies all three inequalities.
Based on the evaluations, the point $\left(x,y\right)=\left(2,1\right)$ and $\left(x,y\right)=\left(0,-1\right)$ are part of the solution set to the given system of inequalities.

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