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| ▭\:\longdivision{▭} | \times \twostack{▭}{▭} | + \twostack{▭}{▭} | - \twostack{▭}{▭} | \left( | \right) | \times | \square\frac{\square}{\square} |
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| - \twostack{▭}{▭} | \lt | 7 | 8 | 9 | \div | AC |
| + \twostack{▭}{▭} | \gt | 4 | 5 | 6 | \times | \square\frac{\square}{\square} |
| \times \twostack{▭}{▭} | \left( | 1 | 2 | 3 | - | x |
| ▭\:\longdivision{▭} | \right) | . | 0 | = | + | y |
Every day, you make decisions with boundaries. You might choose a route to avoid traffic, stay within a budget, or make sure a recipe uses no more than three eggs. These choices all rely on inequalities, which are math statements using symbols like $<$, $\leq$, $>$, and $\geq$ to set limits or compare values. Inequalities help us navigate real life by describing what is possible and what is not. In this article, we will cover what inequalities are, how to solve them, and how Symbolab’s Inequalities Calculator can make each step clearer.
At its core, an inequality is a way of comparing two values. It tells you whether one thing is bigger, smaller, at least as much, or no more than something else. Inequalities use four main symbols: $<$ means “less than,” $>$ means “greater than,” $\leq$ means “less than or equal to,” and $\geq$ means “greater than or equal to.”
Think of a family sharing a pizza. If you want everyone to have a fair slice, you might say, “No one should take more than two pieces.” That includes 0, 1, or exactly 2 slices, anything 3 or higher breaks the rule. In math, that’s $x \leq 2$, where $x$ stands for the number of slices someone takes. Or maybe your bus leaves in less than ten minutes. That’s $t < 10$, where $t$ is the time you have left.
Equations ask for balance and equality, like a set of perfectly even scales. Inequalities, though, are more like open hands, leaving room for a range of answers. When you see $y > 5$, you know $y$ could be $6$, $10$, $100$, or anything bigger than $5$. If the symbol were $y \ge 5$, the single value 5 itself would also be allowed. It describes possibility, not just one answer. Inequalities show up in all sorts of everyday choices, whether you are stretching your budget, checking if you made the team, or hoping your phone battery lasts at least until you get home. Each time, an inequality marks the line between what is allowed and what is not.
It helps to remember that not all math statements set boundaries. Some ask for perfect balance. An equality is a statement where two things must be exactly the same, marked by the $=$ sign. For example, $2x + 3 = 11$ asks, “What value of $x$ makes both sides equal, with no room for more or less?” Equalities are about a single, exact solution.
An inequality is different. It allows for a whole range of possibilities, not just one answer. The symbols $<$ and $>$ mean strictly less-than and greater-than, while $\leq$ and $\geq$ mean less-than-or-equal-to and greater-than-or-equal-to, respectively.Inequalities give you space, while equalities require precision.
When you solve equations (equalities), you look for that one number that works. When you solve inequalities, you are often describing a set—a stretch of numbers or possibilities that all fit the condition.
Not all inequalities look or behave the same. Just as some decisions are quick and simple—like “Is there enough milk for cereal?”—while others are layered with possibilities, inequalities come in a few different forms. Each type is a way of comparing, setting limits, or exploring possibilities.
This is the kind you meet first, and maybe most often. It compares a single value to a number or another expression.
For example:
$2x + 3 < 7$
Here, $x$ might be the number of hours you can spend online before finishing homework. You want $2x + 3$ (your total “screen time” plus other activities) to stay less than $7$ hours. The solution tells you what values $x$ can have.
These use two variables, often $x$ and $y$, and are a way to compare pairs of values.
For example:
$y \geq 0.5x + 1$
Picture planning a fundraiser, where $x$ is the number of tickets sold, and $y$ is the money raised. This inequality helps you see all combinations of tickets and money that meet your goal. These are often shown as shaded regions on a graph, not just points.
Sometimes, the relationship includes squares or higher powers.
For example:
$x^2 - 4x + 3 > 0$
Maybe you are designing a garden, and this inequality describes the safe range for length and width so nothing is overcrowded. Solutions to these can be a bit more complex, and often come in intervals, not single numbers.
Life is rarely about a single limit. Sometimes you need to stay between two boundaries.
For example:
$5 < x \leq 12$
This could be your safe temperature range for a pet fish tank, where $x$ is the temperature in degrees Celsius. Your fish need water warmer than $5$, but no hotter than $12$. Water at exactly 5 °C is too cold (open circle), but exactly 12 °C is still safe (closed circle), so any temperature between them—up to and including 12—keeps the fish happy. Both boundaries matter, and together, they describe a safe zone.
These involve distances from zero, or how far a value can stray from a target.
For example:
$|x - 4| < 2$
Maybe you are trying to keep your running time within $2$ minutes of a $4$-minute goal. The absolute value marks a window of possibility—never too slow, never too fast.
Each type of inequality tells its own story. Some set a single boundary. Some describe a safe or allowed region. Some mark a whole range of possible answers.
Whenever you meet an inequality, ask yourself: what is this really describing? What boundary, what limit, what “safe zone” is being marked? Math may look abstract, but its heart is always practical.
Solving inequalities is a lot like moving furniture. Each step you take changes the room a little, and you keep making small shifts until everything is in its right place. Solving an inequality means working toward getting your variable on one side by itself. Along the way, you follow certain rules that keep the comparison fair.
Look at what you have. For example,
$2x + 5 < 11$
This asks, for which values of $x$ is $2x + 5$ smaller than $11$?
Start by removing constants from the variable side. Here, you subtract $5$ from both sides:
$2x+5−5<11−5$
$2x<6$
Now, divide both sides by $2$ to finish isolating $x$:
$\frac{2x}{2} < \frac{6}{2}$
$x<3$
So $x$ can be any number less than $3$.
Here is where many people trip. If you multiply or divide both sides of an inequality by a negative number, you must flip the inequality symbol. This keeps the comparison true.
For example,
$-3x > 9$
Divide both sides by $-3$. Do not forget to flip the $>$ to $<$:
$\frac{-3x}{-3} < \frac{9}{-3}$
$x<−3$
If you do not flip the sign, your answer will not be correct.
Test your answer with a number that fits the solution and one that does not.
For $x < 3$, try $x = 2$:
$2(2) + 5 = 9$, which is less than $11$, so it fits.
Try $x = 4$:
$2(4) + 5 = 13$, which is not less than $11$. That is outside the solution.
You can express your solution in several ways:
As an inequality: $x < 3$
On a number line: Draw a line, make an open circle at $3$, and shade everything to the left
In interval notation: $(-\infty, 3)$
These all describe every possible value $x$ can have to make the original statement true.
Sometimes an inequality sets two boundaries at once, like
$2 < x \leq 5$
This means $x$ is bigger than $2$ but not more than $5$. It is like being told to keep your phone charged above $20$ percent but not to go over $100$ percent. You have a safe range.
When you see something like $|x - 4| < 2$, it is asking, how far can $x$ be from $4$ without going outside $2$ units? Notice we used the strict “<” symbol, so the endpoints $2$ and $6$ are excluded; if the original statement were $\lvert x-4\rvert \le 2$, the solution would be $2 \le x \le 6$ and the endpoints would be included.
This splits into two inequalities:
$−2<x−4<2$
Add $4$ everywhere to get
$2<x<6$
So $x$ is between $2$ and $6$.
No matter the form, each solution is a guide for what is possible, safe, or allowed. Practice a few by hand so you feel the logic behind every step, not just the answer at the end.
It is easy to make small missteps with inequalities, even when you are working carefully. Here are some of the most common pitfalls to watch for as you solve:
If you want to see how an inequality is solved step by step, the Symbolab Inequalities Calculator can guide you through the process in a clear and supportive way. Here’s how to get started:
After entering your expression, click “Go.”
If anything is unclear or you want a more detailed explanation, use the ‘Chat with Symbo’ feature to ask questions about any step. This approach allows you to work through inequalities at your own speed, building understanding and confidence along the way.
Inequalities appear in everyday decisions, helping you manage spending, time, and safety. This article explains what inequalities are, how to solve them step by step, and how the Symbolab Inequalities Calculator can support your learning. With clear explanations, real-life examples, and steady guidance, you can build confidence and skill in working with inequalities, both in math and in daily life.
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