Always True, Never True
Part of the Inequalities & Absolute Value section of Coddy's Algebra journey. Lesson 25 of 64.
Sometimes the x-terms cancel out. In x + 1 < x + 2, subtract x from both sides. What is left is 1 < 2, with no x in it. The statement 1 < 2 is true, and it stays true whatever value x has. So every number is a solution. The solution set is all real numbers.
In x > x + 1, subtract x from both sides. What is left is 0 > 1, which is false whatever value x has. So no number is a solution.
When the x-terms cancel, look at the statement that is left. If it is true, every number is a solution. If it is false, no number is a solution. Neither result means you made a mistake. Any number is always less than itself plus 1, and never greater than itself plus 1.
Challenge
MediumSolve the inequality 2(x + 3) >= 2x + 5 for x.
Try it yourself
Solve for x
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