💡 Inequalities vs. Equations
Inequalities are like equations with one key difference — instead of an equal sign, you have <, >, ≤, or ≥. You solve them the same way (inverse operations), with one special rule: if you multiply or divide by a negative number, flip the inequality sign!
| Key Word | Symbol | Example |
|---|---|---|
| at most, maximum, no more than, up to | ≤ | budget ≤ $50 |
| at least, minimum, no less than | ≥ | hours ≥ 8 |
| less than, fewer than, below, under | < | speed < 65 |
| more than, greater than, above, over | > | profit > $0 |
The inequality 0.1116x + 23.77 ≤ 50 uses "less than or equal to" because Robert cannot spend MORE than $50 — he can spend exactly $50 (at most $50). The word "without going over" → ≤
Solve inequalities exactly like equations — use inverse operations (add, subtract, multiply, divide) to isolate the variable.
ONE EXCEPTION: If you multiply or divide both sides by a negative number, you must FLIP the inequality sign!
Example: −2x ≤ 10 → divide by −2 → x ≥ −5 (sign flipped!)
Note whether it's <, >, ≤, or ≥ — you'll keep this symbol throughout (unless you divide by a negative).
Same as equations: undo addition/subtraction first, then multiplication/division. Apply each operation to BOTH sides.
Did you divide or multiply by a negative? If yes → flip the symbol. If no → keep it the same.
The variable represents something real. If it asks for "maximum kilowatt hours," x ≤ 235 means the answer is 235.
2x ≤ 10
Divide by +2 → sign stays:
x ≤ 5
−2x ≤ 10
Divide by −2 → sign flips:
x ≥ −5
0.1116x = cost of electricity (rate × kilowatt hours)
23.77 = fixed water and sewer cost
≤ 50 = total must not exceed $50 budget
x = kilowatt hours used — this is what we solve for
0.1116x ≤ 50 − 23.77
0.1116x ≤ 26.23
x ≤ 26.23 ÷ 0.1116
x ≤ 235.04...
x ≤ 235
| Choice | How You'd Get It | Correct? |
|---|---|---|
| 235 | 26.23 ÷ 0.1116 = 235.04 → round down to 235 ✅ | ✅ CORRECT |
| 661 | 50 ÷ 0.1116 = 448... or forgot to subtract 23.77 first ❌ | ❌ Trap! |
| 448 | Used total $50 ÷ 0.1116 — forgot to subtract water/sewer ❌ | ❌ Trap! |
| 424 | Wrong arithmetic at some step ❌ | ❌ Wrong |
x ≤ 235.04 means Robert can use at most 235.04 kWh. Since we must stay at or under budget and are rounding to whole kilowatt hours, 235 keeps him under budget. Using 236 kWh would cost 236 × $0.1116 + $23.77 = $50.09 — over budget!
Used with ≤ or ≥. The boundary value IS included in the solution. "x ≤ 235" — 235 itself is allowed.
Used with < or >. The boundary value is NOT included. "x < 235" — 235 itself is not allowed.
| Inequality | Dot | Arrow Direction | Meaning |
|---|---|---|---|
| x < 5 | ○ open | ← left | all values below 5 |
| x ≤ 5 | ● filled | ← left | 5 and all values below |
| x > 5 | ○ open | → right | all values above 5 |
| x ≥ 5 | ● filled | → right | 5 and all values above |
a = 0.1116, b = 23.77, c = 50, symbol ≤ → should give x ≤ 235.04 → maximum 235 kWh ✅
"Without going over," "at most," "no more than," "maximum" → use ≤
"At least," "no less than," "minimum" → use ≥
In Robert's problem, $23.77 water/sewer is a fixed cost. Subtract it from the budget BEFORE dividing. 448 is the trap answer for skipping this step!
"Maximum" problems: round DOWN (to stay within budget).
"Minimum" problems: round UP (to meet the requirement).
0.1116 is positive, so when you divide both sides by it, the ≤ stays ≤. No flip needed here!
Plug x = 235 back in: 0.1116(235) + 23.77 = 26.226 + 23.77 = $49.996 ≤ $50 ✅
Plug x = 236: 0.1116(236) + 23.77 = $50.11 — over budget! ✅ confirms 235 is correct.
Substitute your answer back into the original inequality. If the inequality holds true, your answer is correct. This takes 30 seconds and can save you from a trap answer!