➕ Algebra 1 · Inequalities
Solving Inequalities: fun ways and memory help
The same idea, the fun way
Pick the one that clicks. Each one explains the same concept from a different angle.
A promise to a friend
Your friend says, “I’ll be there in no more than 15 minutes.” If your friend arrives in exactly 15 minutes, the promise is kept. Now suppose the promise had been “in fewer than 15 minutes.” Arriving in exactly 15 minutes would break it. Let t be the minutes until your friend arrives. The first promise is t ≤ 15 and the second is t < 15.
How it connects: The two promises differ only at the boundary: t ≤ 15 has a closed circle at 15, and t < 15 has an open circle. The limit: time cannot be negative, so only t ≥ 0 makes sense, and a friend who arrives at 15 minutes and 1 second has broken both promises.
The qualifying time
In a made-up track meet, a runner reaches the final if the 400-meter time is at most 52 seconds. Let t be the time in seconds. The rule is t ≤ 52. A runner at exactly 52 seconds reaches the final. A runner at 52.4 seconds does not. A runner at 49 seconds does, because 49 is less than 52.
How it connects: “At most” means ≤: a closed circle at 52 and shading to the left, toward smaller numbers. It can feel backwards that smaller times are better, but the inequality only compares numbers: 49 is less than 52, so 49 is in the shaded part. The limit: a time cannot be 0 or negative, so only part of the shaded ray makes sense in this story.
Coins for the shield
In a made-up game, a shield costs 100 coins. You can buy it when your number of coins c satisfies c ≥ 100. With exactly 100 coins you can buy it. With 99 you cannot. With 250 you can, and you have coins left over.
How it connects: c ≥ 100 is a closed circle at 100 with shading to the right: 100 counts, and so does everything above it. The limit: coins are whole numbers, so the real picture is a row of separate dots (100, 101, 102, and so on), not a continuous shaded line. On a number line for real numbers, we shade the whole ray.
The lake mirror
A calm lake reflects a mountain upside down. Take the waterline as 0 and measure heights in units above it. A ridge 2 units high and a peak 5 units high satisfy 2 < 5. In the reflection, their tips are 2 and 5 units below the waterline, at −2 and −5. Now −2 > −5: the taller peak has the lower tip.
How it connects: A reflection across the waterline is multiplication by −1, and it swaps the order of any two heights. That is why the inequality sign reverses when you multiply or divide both sides by a negative number. The limit: a real reflection only multiplies by −1. Multiplying by −3 also stretches distances, but the order swaps just the same. Adding or subtracting only slides everything up or down, so the order and the sign stay.
The sky’s the limit
The idiom “the sky’s the limit” means there is no limit to what is possible.
How it connects: The shaded part of x > 3 has no end: 4, 400 and 4 million all work, so the shading runs off the number line with an arrow. The limit of the idiom: it fits one direction only. x > 3 is unlimited upward, but it has a boundary at 3 on the other side, and 3 itself is not a solution.
Measure twice, cut once
The proverb “Measure twice, cut once” says to check carefully before you do something you cannot undo, like cutting a board.
How it connects: Check your solution before you trust it: put one number from the shaded part, one from the unshaded part, and the boundary into the original inequality. The limit: a board has one cut, but an inequality has infinitely many solutions. A few tests can catch a mistake like a forgotten sign reversal, but they cannot prove the whole answer is right. Correct steps are what guarantee that.
The dizzy sign
Why did the less-than sign get dizzy? Someone divided both sides by −2 and spun it around, and it landed facing the other way: >.
How it connects: Multiplying or dividing both sides by a negative number reverses the sign: < becomes >. The spinning happens only for that move. Adding, subtracting, or multiplying or dividing by a positive number leaves the sign alone.
Memory help
FLIP
When you multiply or divide both sides by a number, ask: Find the number you multiply or divide by. Look at its sign. If it is negative, Put in the reversed sign (< becomes >, ≤ becomes ≥). If it is positive, keep the sign. Adding and subtracting never reverse anything.
Turn it around
“Negative times, negative divide: turn the sign around with pride.” Only multiplying or dividing both sides by a negative number does that. For adding, subtracting, or multiplying or dividing by a positive number, leave the sign alone.
The hungry alligator
Picture an alligator whose open mouth faces the bigger number. In 5 > 2 and in 2 < 5, the mouth opens toward the 5. The line under ≤ and ≥ is half of an equals sign, which means “or equal”. Limit: the alligator tells you which number is bigger. It does not tell you which way to shade unless x is on the left.
A line under it means a filled circle
The symbols ≤ and ≥ have a line under them, like half of an equals sign. “Or equal” means the boundary counts, so the circle is filled. The symbols < and > have no line, so the boundary does not count and the circle is open (empty).
A negative in the problem is not a flip signal
Only multiplying or dividing both sides by a negative number reverses the sign. In x − 3 > −8 there are two negatives in sight, but you add 3 to both sides and get x > −5 with the sign unchanged.
In, out and edge
To check an answer, send three scouts into the original inequality: one from inside the shaded part (it should be true), one from outside (it should be false), and one from the edge, the boundary (both sides should be equal, and it counts only if the sign has a line under it).