Inequality
In short: Inequality is a statement that compares two values that need not be equal, using the symbols <, >, ≤ or ≥, such as x > 5, which has infinitely many solutions.
6th GradeAlgebra6.EE.B.8
What is Inequality?
An inequality says one side is larger or smaller than the other, instead of exactly equal. A sign reading “riders must be at least 48 inches tall” allows 48, 49, 63.5 and countless other heights.
Precisely, the symbols are \(<\) less than, \(>\) greater than, \(\le\) at most and \(\ge\) at least. A solution is any number that makes the statement true, and the full set of solutions is usually drawn on a number line with an open or closed circle.
How to spot it
- A story uses “at least”, “at most”, “more than”, “fewer than” or “no more than”.
- A limit is stated: a speed cap, a budget, a minimum score or a weight capacity.
- A question asks which numbers satisfy a condition rather than for just one answer.
- A number line has a ray with an open or closed circle at the starting value.
The method
- Translate the words into a symbol: “at least 48” becomes \(h \ge 48\).
- Test a number in the statement to see whether it is a solution.
- Graph on a number line: a closed circle when the endpoint counts (\(\le\), \(\ge\)), an open circle when it does not (\(<\), \(>\)).
- Shade toward the values that make the statement true.
Inequality: solved example
Problem
A bounce house at a Dallas fair holds at most 450 pounds. Write an inequality for the combined weight w of the children inside, and decide whether 392 pounds and 461 pounds are allowed.
Solution
“At most 450” gives \(w \le 450\).
Test 392: \(392 \le 450\) is true. Test 461: \(461 \le 450\) is false. So 392 pounds is allowed and 461 pounds is not.
Try it
Question 1
Is 7 a solution of \(3x – 4 > 15\)?
Show the answer
\(3 \cdot 7 – 4 = 17\), and \(17 > 15\) is true, so yes.
Question 2
Write an inequality for “a number n is fewer than 12” and name the largest whole-number solution.
Show the answer
\(n < 12\); the largest whole number is 11.
Inequality vs. Equation
An equation pins down exact values with an equals sign, and a one-step equation like \(x + 3 = 9\) has one solution. An inequality compares with a sign like \(>\) and normally has a whole range of solutions drawn as a ray.
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Related words
Whole dictionary: A–Z · 6th Grade · Algebra