Represent strict inequalities and test their solutions

Learning goal: Write inequalities of the forms x > c and x < c for constraints, test candidate values and represent their solution sets with number-line rays. Explain why a strict boundary is excluded and why a rational-number interval contains infinitely many solutions.

A model experiment requires a cord longer than 2.5 meters. If x is the cord length in meters, write x > 2.5. Longer than excludes exactly 2.5, so x = 2.5 is not a solution. Lengths 2.6, 3 and 4.25 satisfy the condition. A length of 2.4 does not.

Draw a number line increasing to the right, mark an open circle at 2.5 and shade or draw a ray rightward. The open circle indicates that the boundary value is not included. The ray points toward all greater values and continues beyond the displayed portion. It is not a single arrow instructing the reader to stop at the next integer.

For a temperature reading below −1°C in a fictional observation, write t < −1. The graph has an open circle at −1 and extends left. A reading of −2 satisfies the condition; zero does not. The sign of the boundary does not change the meaning of less than: values farther left are still smaller.

Test a specified set by substitution. For x > 2.5 and candidates {2, 2.5, 2.75, 4}, only 2.75 and four make true statements. There can be more than one solution to an inequality. The candidate set lists values to check; it does not necessarily list every solution to the original inequality. Substitution works the same way for an equation, as in the companion equation lesson, where 5.25 makes x + 2.75 = 8 true and 5 does not.

There are infinitely many rational numbers greater than 2.5. Even between 2.5 and 2.6 there are 2.51, 2.501, 2.5001 and indefinitely more possibilities. No finite list captures the entire ray. Likewise, between any two distinct rational solutions in an interval, their midpoint is another rational number within that interval.

Context can restrict the allowed values. If n counts complete cards and n > 2.5, the possible card counts are whole numbers beginning with three. The underlying inequality on a rational-number line has solutions that are not whole numbers, but the context, which counts whole cards, removes those values. Explain both the inequality and the context's allowable set.

Distinguish strict phrases from inclusive ones. Longer than corresponds to >, and less than corresponds to <. At least or no more than includes the boundary and needs an inclusive symbol, which is a different condition from the strict inequalities practiced here. Do not silently translate “at least 2.5” as x > 2.5, because that would wrongly exclude a permitted boundary value.

The diagram gives two separate examples with labeled axes: x > 2.5 extends right from its open boundary, while t < −1 extends left. The number scales and boundary positions are explicit. Each ray represents many points, including fractions and decimals between labeled integers.

An inequality is a statement that can be true or false for a chosen value. The expression x + 2.5 alone is not an inequality because it does not compare two quantities. An equals sign would assert an exact match, while > or < describes an ordering condition. Reading the sentence aloud can help preserve that distinction.

Do not select a graph only by noticing the direction of the printed symbol. Identify the variable, boundary and whether the allowed values are greater or smaller. For example, 2.5 < x expresses the same condition as x > 2.5. Both permit values to the right of 2.5 on an increasing horizontal line.

Worked contextual test: A measurement must be less than four meters, with candidates 3.9, 4 and 4.1. Only 3.9 satisfies m < 4. Four is the boundary and is excluded; 4.1 lies on the wrong side.

Worked correction: A graph for x > 2.5 marks only the points three, four and five. That may show a few whole-number examples, but it omits infinitely many rational solutions and all larger displayed-range values. A continuous ray with an open boundary represents the full rational solution set.

Guided graphing: Write inequalities from strict comparison phrases, test candidate sets and draw number-line rays. Explain the open circle and direction in complete sentences. Compare x > 2.5 with 2.5 < x.

Independent performance: Create one length constraint and one signed-number constraint. For each, give several solutions and nonsolutions, include the boundary in your tests and draw a scaled graph. Explain why the unrestricted rational solution set is infinite, then show how a whole-object context changes allowable values without changing the inequality's comparison rule.