Balance an equation, bound a decision
What changes when you divide an inequality by a negative number?
Entering midway? Doing the same operation to both sides preserves equality. An inequality describes allowed values rather than one answer.
Start with a concrete question
Inspect the linear rule ax+b against a target. Solve both an equality and the related less-than-or-equal constraint.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
Predict which side of the boundary satisfies the inequality when a is negative.
Change one thing. Inspect what happens.
- Solve 2x+3=9 and test the solution by substitution.
- Change a to −2; test an x on each side of the inequality boundary.
- Set a=0 and compare equal versus unequal constants.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Equation as an intersection
Horizontal: x. Vertical: output.
- 1. a x + b (solid line)
- 2. Target (dashed line)
Axes use the displayed ranges.
Read plotted values as a table
| Series | x | output |
|---|---|---|
| a x + b | -10 | -17 |
| a x + b | 10 | 23 |
| Target | -10 | 9 |
| Target | 10 | 9 |
Follow the calculation
- Equation: 2x + 3 = 9. Subtract 3: 2x = 6.
- Divide by 2: x = 3.
- Inequality 2x + 3 ≤ 9: x ≤ boundary 3.
- Check an example on each side of the boundary.
An equation identifies equal outputs; an inequality identifies a set of allowed inputs. A zero coefficient needs a separate case.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Balance an equation, bound a decision
Recap for entering here
Doing the same operation to both sides preserves equality. An inequality describes allowed values rather than one answer.
The concrete question
Inspect the linear rule ax+b against a target. Solve both an equality and the related less-than-or-equal constraint.
Predict which side of the boundary satisfies the inequality when a is negative.
Work through the mechanism
Subtracting b isolates ax. If a is nonzero, dividing by a gives x=(target−b)/a.
For 2x+3=9, subtract 3 to get 2x=6, then divide by 2 to obtain x=3. For 2x+3≤9, all x≤3 work.
With −2x+3≤9, division reverses the sign and gives x≥−3. Testing x=0 confirms it satisfies the original inequality.
Formula and symbols
For a≠0, equality ax+b=t gives x=(t−b)/a. The inequality direction reverses when dividing by a less than 0.
Test the idea
Solve 2x+3=9 and test the solution by substitution.
Change a to −2; test an x on each side of the inequality boundary.
Set a=0 and compare equal versus unequal constants.
Assumptions and boundaries
One variable and a linear rule only.
The plotted window is finite; equations and allowed sets may extend outside it.
Extend your investigation
High school: Translate a fictional spending limit into an inequality.
College: Express solution sets with interval notation and prove the negative-division rule.
Working professionals: Write eligibility thresholds with explicit inclusive boundaries and test cases.
Try a fresh case
Solve −3x+2≤11. What is the lower boundary of the allowed x values?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Translate a fictional spending limit into an inequality.
College investigation
Express solution sets with interval notation and prove the negative-division rule.
Workplace investigation
Write eligibility thresholds with explicit inclusive boundaries and test cases.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.