Same Math › Algebra 1 · grades 8–10 › Graphing inequalities
How do kids solve and graph inequalities?
Inequalities are solved like equations, with one extra rule: multiplying or dividing both sides by a negative number flips the sign. Then the answer is graphed on a number line: an open circle for < or > (the number isn't included), a closed circle for ≤ or ≥, and shading toward the numbers that work.
- Open circle: < or >. The number itself doesn't count.
- Closed circle: ≤ or ≥. The number counts.
- Negative × or ÷: flip the sign.
Why schools teach it
Answers that are ranges, not single numbers
Seventh graders solve and graph inequalities in context (7.EE.B.4b), and Algebra 1 solves them with variables on both sides (HSA.REI.B.3). Real limits are inequalities: at least 13 years old, no more than $50.
The flip rule is the main trap. Kids test it with numbers: 2 < 3, but −2 > −3, so multiplying by −1 reverses the order.
At the kitchen table
What to say, and what to skip
Say this
Solve it like an equation. Did you multiply or divide by a negative?
Then flip the sign.Is the number itself included?
Open or closed circle?Pick a number in your shading. Does it work?
Which way should you shade?
Toward the numbers that make it true.
Skip this
- Flipping the sign when adding or subtracting a negative.Only multiplying or dividing by a negative flips it.
- Shading by the direction of the arrow symbol.−3 > x means x < −3: shade left. Test a number.
- Writing x = 4 as the answer.The answer is x > 4, a whole range.
See it in action
Solving −2x + 5 ≥ 11
Subtract 5
−2x ≥ 6Divide by −2 and flip the sign
x ≤ −3Graph it
closed circle at −3, shade left
Check: x = −4 gives 8 + 5 = 13 ≥ 11 ✓.
Free practice sheet
Make an inequalities sheet
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Parents also ask
Why does the sign flip?
Multiplying by a negative reverses order on the number line: 2 < 5, but −2 > −5.
What if x is on the right, like 4 < x?
Read it from x's side: x > 4. Or flip the whole thing around.
What's a compound inequality?
Two at once, like −2 < x ≤ 5: a segment between −2 and 5.
How do you graph inequalities with two variables?
Graph the line, dashed for < or > and solid for ≤ or ≥, then shade a side. That comes later in Algebra 1.