Compound Inequality Calculator

This page is designed for compound-inequality intent rather than generic inequality traffic. That distinction matters because students rarely struggle with the symbol itself. They struggle with the logic word in the middle. The difference between AND and OR changes the entire shape of the answer, the number-line shading, the interval notation, and even the language you use to explain why the answer works.

You can work in two ways. If you already know the full problem, paste it directly as x > 1 and x < 5, x < -2 or x > 4, or -3 <= 2x + 1 < 7. If you want more structure, switch modes and enter each branch separately. AND mode uses two linked inequalities that must both be true. OR mode uses two linked inequalities where either branch can be true. Three-Part mode keeps the shared middle expression and lets you operate on all three parts at once.

Once the input is recognized, the experience follows a clean teaching order. First identify the type. Next solve each branch or the three-part chain. Then read the final overlap or union on the dual number line. After that, switch to interval notation and verification so the answer is no longer abstract. The goal is not only to produce a correct interval. The goal is to make the logic visible enough that the next compound inequality feels familiar instead of arbitrary.

A compound inequality combines two comparisons into one statement. The connection can be logical, as in x > 1 and x < 5 or x < -2 or x > 4, or it can be written as a chained statement such as -3 <= 2x + 1 < 7. In every case, the final solution is a set of real numbers, not a single isolated answer. That set may be a bounded interval, a union of two intervals, all real numbers, no solution, or a single point.

There are two main families. AND compound inequalities require both conditions to hold at the same time. That makes the final answer the intersection of the two solution sets. OR compound inequalities require at least one condition to hold. That makes the final answer the union of the two solution sets. A three-part inequality is best understood as an AND statement with a shared middle expression, which is why it often produces a bounded interval after the variable is isolated.

Compound inequalities matter because real problems often create more than one restriction at once. A score may need to stay above a minimum and below a maximum. A measurement may need to be outside a forbidden region. A formula may create a left boundary and a right boundary in the same sentence. If a student does not recognize whether the answer should narrow inward or open outward, even correct algebra can end in the wrong graph. That is why this page treats the logic word as a first-class concept instead of a side note.

AND means both inequalities must be true for the same x-value. In set language, that is an intersection. If one branch says x > 1 and the other says x < 5, the final answer is the overlap between those two sets. Values such as 2, 3, and 4 survive because they satisfy both conditions. Values such as 0 fail the first condition. Values such as 6 fail the second. The overlap is the only part that matters.

On a number line, AND usually narrows the solution. That is why teachers often describe it as the middle-interval pattern. When the branches point toward each other, the answer is usually a bounded interval such as (1, 5) or [-2, 3). When both branches point right, the stricter lower bound wins. When both branches point left, the stricter upper bound wins. When the two branches never meet, the answer is no solution because there is no shared region at all.

A useful mental model is constraint stacking. Each new AND condition removes values that fail the new rule. The solution set can only stay the same size or get smaller. It cannot grow. That is why AND problems often produce the tighter bound, the smaller interval, or the empty set. If you remember that AND narrows the surviving set, the final graph becomes easier to predict before you even finish the algebra.

OR means at least one inequality must be true. In set language, that is a union. If one branch says x < -2 and the other says x > 4, the final answer includes both surviving regions because a value is allowed to satisfy either branch. A number such as -5 works because it satisfies the left inequality. A number such as 7 works because it satisfies the right inequality. The fact that neither number satisfies both conditions does not matter. OR never requires both to be true at the same time.

On a number line, OR usually opens the solution outward. That is why many OR answers look like two rays extending toward negative infinity and positive infinity. If the two branches overlap or even touch, the union can merge into one larger interval. If the branches cover the full number line together, the answer becomes all real numbers. In other words, OR is not always two separated pieces, but it almost always keeps more values than either branch alone.

A useful memory line is that OR widens instead of narrows. Each new OR condition gives x another way to survive. The solution set can only stay the same size or get larger. It cannot shrink. That is why OR problems often produce the looser bound, the larger interval, or the entire real line. If you think of OR as opening doors rather than closing them, the final interval notation becomes much more intuitive.

The first solving decision is structural, not algebraic. Before moving any terms, identify whether the problem is AND, OR, or three-part. That choice tells you whether the final operation will be an intersection, a union, or a simultaneous chain simplification. Skipping this step is one of the most common reasons students turn a correct branch solution into the wrong final interval.

For AND and OR problems, the safest workflow is to solve each inequality separately first. Rewrite each branch until x is isolated, convert each branch into interval notation, and then combine the intervals with the correct set operation. For AND, keep only the overlap. For OR, keep every region that survives at least one branch. If the intervals are adjacent or overlapping in an OR problem, combine them into a single interval before writing the final answer.

For three-part inequalities, the best workflow is different. Because the middle expression is shared, you can often add, subtract, multiply, or divide all three parts at the same time. This is faster and cleaner than splitting the chain immediately. The one rule that still deserves extra attention is the sign-flip rule: if you multiply or divide all three parts by a negative number, both inequality signs reverse direction. After that, write the final condition in standard order and convert it into interval notation.

A three-part compound inequality has the form a [sign] expression [sign] b. A common example is -3 <= 2x + 1 < 7. This format is compact, but it is not mysterious. It is equivalent to two inequalities at once: -3 <= 2x + 1 and 2x + 1 < 7. Because the middle expression is shared, the final answer comes from the overlap of both comparisons, which is why three-part problems belong to the AND family.

The main advantage of the three-part format is efficiency. If you subtract 1 from the middle expression, you also subtract 1 from both outer parts. If you divide by 2, you divide all three parts by 2. That lets you isolate x without splitting the chain immediately. Students who understand this move gain speed and reduce transcription errors because they keep the structure intact while solving.

The main danger is forgetting that a negative multiplier changes both inequality signs. Consider -4 < -2x + 6 <= 10. After subtracting 6, you get -10 < -2x <= 4. Dividing by -2 gives 5 > x >= -2, and both signs flip. If you miss even one flip, the final interval is wrong. That is why this page treats three-part solving as a visible sequence of operations rather than a one-line answer.

Graphing is where compound-inequality logic becomes obvious. In an AND problem, draw the first branch, draw the second branch, and then keep only the region where the shadings overlap. In an OR problem, draw both branches and keep every highlighted region. The dual number line on this page makes that comparison explicit by showing branch one, branch two, and the final result on separate but synchronized tracks.

This matters because many wrong answers look plausible in algebra form but fall apart on a graph. A student may correctly solve x > 1 and x < 5, then accidentally shade two rays instead of the middle region. Another student may solve x < 8 or x > -1 and fail to notice that those two rays cover the entire number line. The graph acts as a second proof. If the picture does not match the logic word, the final answer deserves another look.

When graphing, pay attention to endpoint style as well as shading direction. Strict inequalities such as < and > use open circles because the endpoint is excluded. Inclusive inequalities such as <= and >= use closed circles because the endpoint is part of the solution. In a compound problem, the final endpoint style depends on the surviving interval after the intersection or union is formed, not only on one branch in isolation.

Interval notation compresses the final answer into a precise form that is easy to scan. A bounded AND answer such as 1 < x < 5 becomes (1, 5). A mixed-endpoint three-part answer such as -2 <= x < 3 becomes [-2, 3). An OR answer such as x < -3 or x > 8 becomes (-infinity, -3) union (8, infinity). The notation is compact, but every bracket and parenthesis still carries meaning.

The fastest way to avoid mistakes is to connect interval notation to the graph. Open circle means parenthesis. Closed circle means bracket. Infinity always uses parentheses because infinity is never an included endpoint. If the solution contains two separated pieces, the notation must use a union symbol. If the graph is empty, the notation is the empty set. If the graph is a single point, set notation such as {3} can be clearer than interval notation.

On this page, the condition form, interval notation, and set notation are displayed together because each representation reinforces the others. The condition form is easiest for beginners to read. Interval notation is shortest for homework and tests. Set notation is useful when you want to state the rule verbally. Switching among the three is not busywork. It is a way to verify that the same mathematical idea survives every notation system.

Some compound inequalities are valuable precisely because they expose edge cases. In an AND problem such as x > 5 and x < 2, the two branches never overlap, so there is no solution. The empty set is not a failure of algebra. It is the correct description of a contradictory requirement. Likewise, x >= 3 and x <= 3 does not produce a wider interval. It collapses to the single point x = 3 because only one value satisfies both bounds at once.

OR problems have their own signature special case: all real numbers. If one branch says x < 8 and the other says x > -1, every real number satisfies at least one branch because the two surviving regions overlap enough to cover the full number line. This is one of the clearest moments where a graph prevents mistakes. The union does not merely create two rays. It erases the gap completely.

Three-part inequalities can also become contradictory. A statement such as 5 < x < 2 asks x to be greater than 5 and less than 2 at the same time. There is no real number that can do that, so the answer is no solution. These edge cases matter because they teach students not to assume every compound inequality must produce a standard interval. A strong solver checks the final set before stopping.