What Is the Solution to the Inequality? The Solution Set, the Boundary, and Three Notations

The roller-coaster sign is the idea in its purest form: at least 48 inches to ride. Height $h$ must satisfy $h\ge 48$. A 48-inch rider passes, a 47-inch rider waits, every inch above 48 passes.

So what is the solution to the inequality? The complete set of values that make it true: every real number from 48 up, written $h\ge 48$. The closed dot says the boundary itself rides; the shading says all beyond it do. Always be prepared to explain how you know.

What is the solution to the inequality $x+3<0$? Subtract 3: $x<-3$. No single number settles it: $-4$, $-3.5$, $-100$ all work. The answer is a set: all reals left of $-3$.

Equation solutions are single values; inequality solutions are infinite intervals. A worksheet that asks what is the solution to the inequality wants the whole set - one sample point is not an answer.

Every inequality solution lives or dies at the boundary where true flips. What is the solution to the inequality near that edge? Read the symbol: a strict sign ($<$, $>$) excludes it - $x<3$ refuses 3. An "or equal to" sign ($\le$, $\ge$) admits it. The lines below differ by that sign.

So the whole verdict half-hangs on one dot: open circle, out; closed dot, in. Quick drill: what extra member does $x\le 3$ have? Only the dot itself.

Solving is half the job; what is the solution to the inequality needs a written form. Once solved, the same set needs a written form: dressed three ways plus a picture.

Read a row backward: given $(-\infty, -3]$, say it in words: all numbers up to $-3$, boundary included. Two guardrails apply: infinity never takes a bracket; mixed endpoints must match. First column: for $x\le -3$, the set is $(-\infty, -3]$. The reply stays fixed: what is the solution to the inequality? Every qualifying value, in full.

Does a given number really belong? The standing question - what is the solution to the inequality? - is settled one number at a time by substitution.

1. Substitute the candidate for the variable.
2. Simplify each side separately.
3. Read the verdict: true admits, false rejects.

Example: $1\le 2$ true - so for $x\le 2$ the solution set is every number at or under 2.

This is where what is the solution to the inequality gets graded: samples in, truth out. Substitute, and let truth vote.

Problem. Decide whether $y=2$ is a solution of $y\le 5$, and of $y>5$.

Test in $y\le 5$. $2\le 5$ - true. So $y=2$ belongs to the solution set - one member of what is the solution to the inequality $y\le 5$.

Test in $y>5$. $2>5$ - false. So $y=2$ is not a solution: one candidate, two verdicts - the sign decides.

One test pair, two verdicts. So, what is the solution to the inequality $y\le 5$? Every number through 5. By contrast, $y>5$ owns all numbers above 5 - 2 fails, 5 fails, 5.001 passes. In interval form: $(-\infty, 5]$.

Problem. Solve $x+4\le 9$; write the answer in interval notation; check the boundary.

Step 1. Subtract 4: $x\le 5$ - subtracting never flips a sign - what is the solution to the inequality after one move? Still $x\le 5$.

Step 2. Interval: $(-\infty, 5]$ - the bracket records 5 itself.

Check. Boundary $5\le 5$: true - the boundary 5 is kept. Outside, $6\le 5$: false - rejected. All three moves leave that ray through 5 and below. Either way, what is the solution to the inequality $x+4\le 9$? A set, never a point.

Problem. Solve $-3x\le 12$; state the interval; check the boundary.

Step 1. Divide by $-3$ - a negative - so the sign flips: $x\ge -4$. Before versus after: for $-3x\le 12$, what is the solution to the inequality? The ray from $-4$ up.

Step 2. Interval: $[-4, \infty)$ - The bracket carries the "or equal to" across the flip, and the answer reads rightward.

Check. Boundary: $12\le 12$, true - kept. Outside: $15\le 12$, false. Without the flip, the answer points the wrong way. For $-3x\le 12$ the solution set is every number from $-4$ upward. The check closes it: the solution set of $-3x\le 12$ is $[-4, \infty)$. (The same mechanism: $-2y<-8$ becoming $y>4$.) And what is the solution to the inequality $-2y<-8$? Exactly $y>4$.](streamdown:incomplete-link)

Ask the question "what is the solution to the inequality" and answer in one breath: every value that makes it true - as inequality, interval, or shaded line. That line is the whole answer.

One last drill: what is the solution to the inequality in a single sentence? The complete set of true-making values - checked, not guessed. Before handing work in, decide the boundary, then substitute one candidate per side.

1. Answering with a single number. "$x=5$" for $x\le 5$ throws away every other member. Write $x\le 5$ and the whole ray answers what is the solution to the inequality - 5 is its edge.

2. Skipping the flip. Dividing $-3x\le 12$ by $-3$ without flipping gives $x\le -4$, where the original mostly fails. Flip on every negative: for $-3x\le 12$, what is the solution to the inequality? $x\ge -4$, not $x\le -4$.

3. Misplacing the boundary. Writing $x>7$ for $x\ge 7$ evicts 7 from its own solution. The question "what is the solution to the inequality" turns on that symbol: "or equal to" means closed dot.

4. Bracketing infinity. $[2, \infty]$ is always wrong: infinity is never reachable, so that side takes a parenthesis.

5. Shading the wrong side. Substitute one easy number per side and shade where truth landed. All five fixes share one discipline: ask what is the solution to the inequality, then let set, boundary, and one substitution answer. [源外补充]