After substituting $y=-4$, solve each resulting inequality step by step to find what $x$ must satisfy in each case.

When solving the first inequality, you will divide by a negative number. Remember that this reverses the inequality sign.

Once you have the range of $x$ that works for each inequality, find their overlap. Then see which answer choices fall in that overlapping range.

Enter the inequalities as they are:

• Type y <= -2x + 6 to graph the first half-plane.
• Type y > -x - 3 (rewriting $x + y > -3$ as $y > -x - 3$) to graph the second half-plane. The overlapping shaded region represents all solutions to the system.

Type y = -4 to draw the horizontal line that all points of the form $(x,-4)$ lie on. Look at where this line passes through the overlapping shaded region to see the allowed $x$-values.

Plot each candidate point by entering them one at a time, for example (−1, -4), (1, -4), (5, -4), (6, -4). Check the graph to see which of these points lies inside the overlapping shaded region of the two inequalities. The corresponding $x$-value is the one that works.

The point has coordinates $(x,-4)$, so in both inequalities we can replace $y$ with $-4$.

The system

becomes

Now we just need to solve these two inequalities for $x$.

Start with

Subtract $6$ from both sides:

Now divide both sides by $-2$. Remember: dividing by a negative flips the inequality sign:

This is the same as $x \le 5$.

So from the first inequality, $x$ must be less than or equal to $5$.

Now solve

Add $4$ to both sides:

So from the second inequality, $x$ must be greater than $1$.

From the first inequality we have $x \le 5$, and from the second we have $x > 1$.

Together, $x$ must satisfy

Now test each answer choice:

• $x=-1$: not greater than $1$.
• $x=1$: not greater than $1$ (it equals $1$).
• $x=5$: greater than $1$ and less than or equal to $5$.
• $x=6$: greater than $5$.

The only choice that fits $1 < x \le 5$ is $x=5$, so $5$ is the correct value of $x$.