Is the Endpoint In?
Part of the Inequalities & Absolute Value section of Coddy's Math journey. Lesson 28 of 64.
The dot is the detail people get wrong. 3x + 1 <= 7 solves to x <= 2, and because the sign is ≤, the number 2 itself is a solution: 3 × 2 + 1 = 7 and 7 is less than or equal to 7. The dot at 2 is filled.
3x + 1 ≤ 7, that is x ≤ 2: a filled dot at 2
A strict sign would give an open dot: 3x + 1 < 7 has the same endpoint, 2, but 2 is not a solution, so the dot is hollow. The sign of the solved inequality decides; nothing else.
Challenge
Show the solutions of 4x - 2 >= 6 on the number line.
Try it yourself
Draw the graph
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Inequalities & Absolute Value
1What an Inequality Is
The Four SignsA Picture on a LineClosed or OpenReading It BackwardsRecap: What an Inequality Is4Two-Step Inequalities
Two StepsSubtract, Then FlipWith a FractionA Bracket on the WayRecap: Two-Step Inequalities2Adding and Subtracting
Add to Both SidesSubtract from Both SidesNegatives in the MixSolve, Then DrawRecap: Adding and Subtracting5Variables on Both Sides
x on Both SidesGather Toward the LargerBrackets FirstAlways True, Never TrueRecap: Variables on Both Sides11From Words to Inequalities
At Least, At MostMore ThanA Budget That ShrinksTwo LimitsRecap: From Words3Multiplying and Dividing
Divide by a PositiveMultiply by a PositiveThe FlipDividing by a NegativeNegative FractionsRecap: Multiply and Divide6Graphing Solutions
From Inequality to PictureIs the Endpoint In?A Flip on the LineRecap: Graphing Solutions