How to Solve One-Step Inequalities?
Read,5 minutes
An inequality compares two expressions with symbols such as \(<\), \(>\), \(≤\), and \(≥\). Solve an inequality much like an equation: isolate the variable by doing the same operation to both sides.
The important exception is multiplication or division by a negative number. When both sides are multiplied or divided by a negative number, reverse the inequality symbol.
Graphing the solution
- Use an open circle for \(<\) or \(>\).
- Use a closed circle for \(≤\) or \(≥\).
- Shade left for less than and right for greater than.
One Step Inequalities
Think of this lesson as more than a rule to memorize. One Step Inequalities is about comparing values and tracking when the sign reverses. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.
An inequality compares sizes. Solve it like an equation, but remember that multiplying or dividing by a negative number reverses the inequality sign.
Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.
- Clear clutter such as parentheses or fractions.
- Collect like terms.
- Undo operations in reverse order.
- Substitute the answer back or test a point.
A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.
Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.
When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.
On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.
Free printable Worksheets
Exercises for One-Step Inequalities
1) Solve and graph \(x + 7 < 3\)
2) Solve and graph \(x - 5 ≥ 9\)
3) Solve and graph \(x + 12 ≤ -1\)
4) Solve and graph \(x - 8 > -2\)
5) Solve and graph \(3x < 18\)
6) Solve and graph \(-4x ≤ 20\)
7) Solve and graph \(x/2 > -6\)
8) Solve and graph \(-x < 11\)
9) Solve and graph \(x + 4 > 10\)
10) Solve and graph \(x - 13 ≤ -20\)
11) Solve and graph \(5x ≥ -25\)
12) Solve and graph \(-2x > 14\)
13) Solve and graph \(x/3 ≤ 4\)
14) Solve and graph \(-x/5 ≥ 2\)
15) Solve and graph \(x + 9 ≥ 9\)
16) Solve and graph \(x - 2 < -9\)
17) Solve and graph \(7x > 0\)
18) Solve and graph \(-3x < -12\)
19) Solve and graph \(x/4 ≥ -3\)
20) Solve and graph \(-6x ≤ -30\)
1)Solve and graph \(x + 7 < 3\)
Subtract \(7\) from both sides: \(x + 7 - 7 < 3 - 7\), so \(x < -4\).
Solution: \(x < -4\). Use an open circle at \(-4\) and shade to the left.
2)Solve and graph \(x - 5 ≥ 9\)
Add \(5\) to both sides: \(x ≥ 14\).
Solution: \(x ≥ 14\). Use a closed circle at \(14\) and shade to the right.
3)Solve and graph \(x + 12 ≤ -1\)
Subtract \(12\) from both sides: \(x ≤ -13\).
Solution: \(x ≤ -13\). Use a closed circle at \(-13\) and shade to the left.
4)Solve and graph \(x - 8 > -2\)
Add \(8\) to both sides: \(x > 6\).
Solution: \(x > 6\). Use an open circle at \(6\) and shade to the right.
5)Solve and graph \(3x < 18\)
Divide by positive \(3\): \(x < 6\).
Solution: \(x < 6\). Use an open circle at \(6\) and shade to the left.
6)Solve and graph \(-4x ≤ 20\)
Divide by negative \(-4\) and reverse: \(x ≥ -5\).
Solution: \(x ≥ -5\). Use a closed circle at \(-5\) and shade to the right.
7)Solve and graph \(x/2 > -6\)
Multiply by positive \(2\): \(x > -12\).
Solution: \(x > -12\). Use an open circle at \(-12\) and shade to the right.
8)Solve and graph \(-x < 11\)
Divide by \(-1\) and reverse: \(x > -11\).
Solution: \(x > -11\). Use an open circle at \(-11\) and shade to the right.
9)Solve and graph \(x + 4 > 10\)
Subtract \(4\): \(x > 6\).
Solution: \(x > 6\). Use an open circle at \(6\) and shade to the right.
10)Solve and graph \(x - 13 ≤ -20\)
Add \(13\): \(x ≤ -7\).
Solution: \(x ≤ -7\). Use a closed circle at \(-7\) and shade to the left.
11)Solve and graph \(5x ≥ -25\)
Divide by \(5\): \(x ≥ -5\).
Solution: \(x ≥ -5\). Use a closed circle at \(-5\) and shade to the right.
12)Solve and graph \(-2x > 14\)
Divide by \(-2\) and reverse: \(x < -7\).
Solution: \(x < -7\). Use an open circle at \(-7\) and shade to the left.
13)Solve and graph \(x/3 ≤ 4\)
Multiply by \(3\): \(x ≤ 12\).
Solution: \(x ≤ 12\). Use a closed circle at \(12\) and shade to the left.
14)Solve and graph \(-x/5 ≥ 2\)
Multiply by \(5\): \(-x ≥ 10\). Divide by \(-1\) and reverse: \(x ≤ -10\).
Solution: \(x ≤ -10\). Use a closed circle at \(-10\) and shade to the left.
15)Solve and graph \(x + 9 ≥ 9\)
Subtract \(9\): \(x ≥ 0\).
Solution: \(x ≥ 0\). Use a closed circle at \(0\) and shade to the right.
16)Solve and graph \(x - 2 < -9\)
Add \(2\): \(x < -7\).
Solution: \(x < -7\). Use an open circle at \(-7\) and shade to the left.
17)Solve and graph \(7x > 0\)
Divide by \(7\): \(x > 0\).
Solution: \(x > 0\). Use an open circle at \(0\) and shade to the right.
18)Solve and graph \(-3x < -12\)
Divide by \(-3\) and reverse: \(x > 4\).
Solution: \(x > 4\). Use an open circle at \(4\) and shade to the right.
19)Solve and graph \(x/4 ≥ -3\)
Multiply by \(4\): \(x ≥ -12\).
Solution: \(x ≥ -12\). Use a closed circle at \(-12\) and shade to the right.
20)Solve and graph \(-6x ≤ -30\)
Divide by \(-6\) and reverse: \(x ≥ 5\).
Solution: \(x ≥ 5\). Use a closed circle at \(5\) and shade to the right.
One Step Inequalities Quiz
