Welcome to Solving One-Step Inequalities! This is the very first lesson of the course, and we are starting with a skill that builds directly on everything you learned in the previous course, Understanding Inequalities. There, you explored what inequality symbols mean, how to check whether a value is a solution, and how to read inequalities on a number line. Now it is time to put that knowledge to work and learn how to solve inequalities on your own. In this lesson, we will focus on solving one-step inequalities using addition and subtraction.
A Quick Refresher Before We Start
Because solving inequalities relies on "undoing" operations, it helps to be comfortable with a couple of ideas from earlier math. Take a moment to make sure these feel familiar:
Inverse operations undo each other. Addition and subtraction are inverses, so to undo "+6" we subtract 6, and to undo "−4" we add 4. This is the same move you used when solving one-step equations like x−5=3.
Signed-number arithmetic. You will often add or subtract with negative numbers, so remember that −1−9=−10 and −2+7.5=5.5. If working with negatives feels shaky, it is worth reviewing that first, since the arithmetic — not the method — is where most mistakes happen.
If both of these ideas make sense, you are ready to solve inequalities. If not, a short review of one-step equations and signed-number arithmetic will make everything that follows much smoother.
From Equations to Inequalities
Solving by Adding to Both Sides
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If you have ever solved a simple equation like x−5=3, you already know the core idea: do the same operation to both sides to get the variable alone. Solving an inequality works almost the same way. The only difference is that instead of an equals sign, we have a symbol like >, <, ≥, or ≤, and our answer describes a range of values rather than a single number.
The good news for this lesson is that when we add or subtract the same number on both sides of an inequality, the direction of the inequality symbol does not change. That makes the process feel just like solving an equation. Let's see exactly how it works.
When a number is being subtracted from the variable, we undo it by adding that same number to both sides. Consider the inequality:
x−4>3
We want x by itself on the left. Since 4 is being subtracted from x, we add 4 to both sides:
x−4+4>3+4x>7
The solution tells us that every value greater than 7 satisfies the original inequality. Notice that the > symbol stayed exactly the same throughout the process.
Solving by Subtracting from Both Sides
Why the Direction Never Changes
Adding or subtracting shifts every point on the number line by the same amount in the same direction. Imagine two people standing on a number line, one to the left of the other. If both take three steps to the right, the person who was farther right is still farther right. Their relative positions have not changed.
A number-line picture can make this easier to see:
This is exactly why the inequality symbol keeps its direction when we add or subtract. We are simply sliding both sides of the inequality by the same distance, so the "bigger" side stays bigger and the "smaller" side stays smaller.
Working with Negatives and Decimals
Solving Checklist
Conclusion and Next Steps
In this lesson, you learned that solving a one-step inequality by adding or subtracting works just like solving an equation: perform the inverse operation on both sides, and the variable is isolated. The key takeaway is that addition and subtraction never change the direction of the inequality symbol, because shifting both sides by the same amount preserves their relative order.
Up next, you will put this skill into action with a set of hands-on practice tasks. These range from a guided fill-in-the-blank walkthrough to solving inequalities with trickier numbers and even a real-world budgeting scenario, so get ready to make these steps second nature!
When a number is being added to the variable, we undo it by subtracting that number from both sides. Imagine you have already loaded 6 songs onto a playlist and the platform allows at most 10. If x represents additional songs, the inequality is:
x+6≤10
Subtract 6 from both sides:
x+6−6≤10−6x≤4
The direction of the symbol (≤) did not change. The solution means you can add 4 more songs, or any number fewer than 4.
The same add-or-subtract rule applies no matter what kinds of numbers appear. The symbol direction stays the same; the only thing we need to watch is the arithmetic itself. Let's look at two quick examples.
Example 1: Solve x+9≥−1.
Subtract 9 from both sides:
x≥−1−9x≥−10
Example 2: Solve x−7.5≥−2.
Add 7.5 to both sides:
x≥−2+7.5x≥5.5
Take your time with the arithmetic in cases like these. Negative numbers and decimals do not change the method — only the computation requires a bit more care.
Whenever you face a one-step add/subtract inequality, follow this short checklist:
Identify what is being added to or subtracted from the variable.
Apply the inverse operation to both sides (add if the term is subtracted, subtract if it is added).
Simplify both sides.
Keep the inequality direction unchanged.
The table below collects every example from this lesson so you can see the pattern at a glance:
