One-variable equations & inequalities
The idea
An equation is a balance claim: 30 + 11w = 96 insists that both sides are the same number, even though one side hides an unknown. Solving means finding the value of the unknown that makes the claim true, and the method grows straight out of the properties of operations you already trust: whatever you do to one side, do to the other, and the balance survives. Inequalities work the same way but claim 'at least' or 'less than' instead of 'equal', so their answers are whole ranges.
Think of the unknown as wrapped in layers of operations, and solve by unwrapping in reverse order — undo the addition before the multiplication, just as you take off shoes before socks because removing socks first would fail. The classic mistake is operating on only one side, or 'moving' a term across the equals sign without flipping what it does. One inequality-specific care point: multiplying or dividing both sides by a negative number reverses the direction of the inequality sign.
Worked example
A skateboard costs $96. You already have $30 saved and you put away $11 every week. After how many weeks can you buy it?
- Model the savings: after w weeks you hold the starting $30 plus $11 per week, so the equation is 30 + 11w = 96.
- Undo the addition first by subtracting 30 from both sides: 11w = 96 − 30 = 66.
- Undo the multiplication by dividing both sides by 11: w = 66 ÷ 11 = 6.
- Check the solution in the original equation: 30 + 11 × 6 = 30 + 66 = 96, so the balance holds and 6 weeks is exactly enough.
Answer. You can buy the skateboard after 6 weeks of saving.
Check your understanding
- Why must every operation be applied to both sides of an equation, and what breaks if it is not?
- How do you decide which operation to undo first when the unknown is wrapped in several?
- What would the setup and answer look like if the question asked when your savings exceed $96 instead of equal it?
- Why does multiplying an inequality by a negative number flip its direction — can you find a small example that shows it?
Build the foundations first
One-variable equations & inequalities builds on these concepts. If any feel shaky, start there.