Solving inequalities using options

For some inequality questions, plugging in the answer choices is faster than solving.
You substitute each choice into the inequality and check whether it makes the statement true.
The choices that work are your answers, and the ones that fail are eliminated.

Suppose we want a value that satisfies $x < 3$, and the choices are $2$, $4$, and $5$.
Test $2$: is $2 < 3$? Yes, so $2$ works.
Test $4$: is $4 < 3$? No, so eliminate it.
Test $5$: is $5 < 3$? No, so eliminate it.

Often you do not need to test every value — just the edge cases.
These are the boundary value, $0$, or a negative number, and a wrong choice usually breaks at one of them.
Say you are told $x > 5$ and asked which must be true, $x > 3$ or $x > 7$.
Test a value just past the edge, like $x = 6$: it fits $x > 5$, it makes $x > 3$ true, but it makes $x > 7$ false.
That one edge value is enough to eliminate $x > 7$.

The boundary value itself is worth extra care.
For a strict inequality like $x > 48$, the edge value $48$ does not count, because $48$ is not greater than $48$.
For $x \geq 48$ it would count.
Testing that exact edge value quickly rules out any choice that wrongly includes or leaves out the boundary.

The same approach works for a system of inequalities, where the answer choices are points $(x, y)$.
Substitute each point into every inequality in the system.
The correct point satisfies all of them; if a point fails even one inequality, eliminate it.

A roller coaster only allows riders taller than $48$ inches. Three riders are $45$, $48$, and $50$ inches tall. Using the rule $h > 48$, which riders can ride?
Test $45$: is $45 > 48$? No.
Test $48$: is $48 > 48$? No — the rule says taller than $48$, and $48$ is not taller.
Test $50$: is $50 > 48$? Yes.
So only the $50$-inch rider can go on.

A student scored more than $80$ on a test, so $s > 80$. A tutor claims two things must be true: $s > 75$ and $s > 85$. Which one must be true?
Test a value just past the edge, like $s = 81$: it satisfies $s > 80$.
Check $s > 75$: $81 > 75$ is true.
Check $s > 85$: $81 > 85$ is false.
The edge value $81$ breaks $s > 85$, so only $s > 75$ must be true.

On a cold day the temperature $t$ is below zero, so $t < 0$. Which expression must also be below zero: $2t$ or $t + 3$?
Test an edge value close to $0$, like $t = -1$:
$2t = 2(-1) = -2$, which is below zero.
$t + 3 = -1 + 3 = 2$, which is not below zero.
The edge value near $0$ breaks $t + 3$, so only $2t$ must be below zero.

Which point is a solution of the system $y > x + 1$ and $y < 5$: $(0, 0)$, $(1, 3)$, $(2, 6)$, or $(4, 2)$?
A solution must satisfy both inequalities, so substitute each point into both.
$(0, 0)$: is $0 > 0 + 1$? No, so eliminate it.
$(1, 3)$: is $3 > 1 + 1$? Yes. Is $3 < 5$? Yes. It works.
$(2, 6)$: is $6 < 5$? No, so eliminate it.
$(4, 2)$: is $2 > 4 + 1$? No, so eliminate it.
Only $(1, 3)$ satisfies both, so it is the solution.