Inequalities: a set of values rather than one answer

Where we are on the map

An equation may identify a point; an inequality describes an entire region of valid values.

Begin with a familiar image

If an elevator carries at most 600 kg, many loads from 0 to 600 are valid rather than one exact weight.

Precise meaning

< and > exclude the boundary; ≤ and ≥ include it. Multiplication by a negative reverses number-line order, so the sign reverses.

The lesson’s support signal

meaning → model → calculation → check → explain in your own words

find boundary → included? → shade line → test one valid and one invalid value

Worked example

3x+2≤14; 3x≤12; x≤4. The boundary 4 works, while 5 does not.

Example with a fading prompt

Change the numbers in the example and repeat the solution without copying its steps. Estimate first, calculate exactly, and check against the original condition.

Recall without a prompt

1. Define the main idea in one sentence.
2. Rebuild the support signal from memory.
3. Solve the example with different numbers and explain why every step is valid.

Find and correct the mistake

Dividing −2x<6 and keeping x<−3 is wrong: division by −2 reverses the order, giving x>−3.

Transfer to a new setting

Apply the same relationship to something you can measure yourself. Name the quantities, units, and the boundary within which the model is valid. Check the answer by a second representation or an inverse operation.

Exercise

Required. Solve 5−2x≥−7, show it on a number line, and test the boundary with two values.

With your own data. Create one everyday example with the same structure and show it in words, a formula, and a check.

Optional. Seven days later, change the numbers and repeat without the support map.

Open perspective

The prealgebra checkpoint combines variables, expressions, equations, coordinates, and inequalities.

Course contents