Lesson 3 of 3
Keep equations balanced and state inequality endpoints precisely.
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Big ideas · 1 of 8
Big ideas
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An equation keeps the same solutions when you add or subtract the same quantity on both sides, or multiply or divide both sides by the same nonzero constant.
Distribute carefully and combine like terms. Moving a term is shorthand for a balancing operation, not a rule that changes signs without reason.
Multiplying or dividing an inequality by a negative number reverses its direction. Adding the same number to both sides does not reverse it.
A bracket includes a finite endpoint; a parenthesis excludes it. Use parentheses with infinity. An equation can have one solution, no solution or every real input as a solution.
Step-through
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Worked example: keep both sides balanced
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Both sides agree.
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Input −3 makes equality; input −4 gives .
Do this: Explain why the tempting shortcut fails.
Wrong idea: dividing by −3 leaves . A negative divisor reverses the order, giving . A test input such as confirms the corrected inequality.
Worked example
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Worked example
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Then , so . The interval is .
Watch
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Flashcards
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Card 1 of 8
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Equality must survive each step.
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What keeps an equation balanced?
Apply the same valid operation to both sides.
Solve .
.
Solve .
.
When does an inequality direction reverse?
When multiplying or dividing both sides by a negative number.
Write in interval notation.
.
Write in interval notation.
.
What does an identity such as mean?
Every real x solves the equation.
What does a contradiction such as mean after valid steps?
The original equation has no solution.
True or false
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Self-test
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6 questions. One at a time.
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Self-test
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6 questions. One at a time.
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Write it
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Write it
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Match
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Connect the ideas
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Homework · 3 points
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Homework
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https://gradebridge-beryl.vercel.app/learn/precalculus/essential-review/linear-equations-and-intervals#homework
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