BetterGrades Precalculus · Unit 1 · Lesson

Equation and inequality repair

Classify and solve common equation and inequality families while distinguishing reversible steps from candidate-generating steps.

Opening

Start with the situation

Equation solving identifies inputs that make a statement true; inequality solving identifies an interval or union of inputs.

These algebra choices are the load-bearing steps beneath later work with functions. A restriction lost here can turn into a false intercept, a missing asymptote, or an invalid model several lessons later.

Before you begin

Prerequisite check

Use signed arithmetic accurately.
Show algebraic steps in a checkable order.
State restrictions before simplifying.

Core explanation

Explanation

Classify the family, state restrictions, use reversible steps when possible, label candidate-generating steps such as squaring or denominator clearing, and check candidates in the original statement.

Multiplying or dividing an inequality by a negative number reverses the order. Rational, radical, logarithmic, and even-power steps require domain checks.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through equation-family decision map, equivalence versus implication ledger, or another equivalent representation.

Conceptual reading

What the idea is really doing

Precalculus is unforgiving about hidden algebra errors. The useful habit is to separate reversible algebra from steps that can create candidates, and to keep domain restrictions beside the work instead of trying to remember them at the end.

This lesson narrows that lens to one goal: classify and solve common equation and inequality families while distinguishing reversible steps from candidate-generating steps. The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.

Reusable method

A reliable route through the problem

Classify the family.
State restrictions.
Use reversible steps when possible.
Label candidate-generating steps such as squaring or denominator clearing.

Verification: Re-read the original statement, not only the simplified line. Confirm every restriction, substitute each candidate, and describe what the result means before moving on.

Foundation walkthrough

Plan before calculating

Problem

Solve $sqrt(x+1)=x-1$

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Classify the family, state restrictions, use reversible steps when possible, label candidate-generating steps such as squaring or denominator clearing, and check candidates in the original statement.
Conclusion: The domain requires $x\ge 1$ . Squaring gives candidates $0$ and $3$ ; only $x=3$ checks.
Why the check works: Squaring can enlarge the candidate set.

Worked examples

See the idea in three forms

foundation example

Solve $sqrt(x+1)=x-1$

SolutionThe domain requires $x\ge 1$ . Squaring gives candidates $0$ and $3$ ; only $x=3$ checks.

Squaring can enlarge the candidate set.

representation example

Solve $|2x-1|=7$

Solution $x=4$ or $-3$ .

This example expresses equation and inequality repair in a second form.

transfer example

Solve $3^(x+1)=81$

Solution $x=3$

Multiplying or dividing an inequality by a negative number reverses the order. Rational, radical, logarithmic, and even-power steps require domain checks.

Equation-family decision map. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Squaring can enlarge the candidate set. — **Anchor figure · Equation-family decision map**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Squaring can enlarge the candidate set.

Equivalence versus implication ledger. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for equation and inequality repair. — **Mechanism figure · Equivalence versus implication ledger**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for equation and inequality repair.

Negative scaling reflection. Compare the valid path with the tempting shortcut. The figure shows why reporting every algebraic candidate as a solution without checking the original equation leads to a false conclusion. — **Comparison and error figure · Negative scaling reflection**
Compare the valid path with the tempting shortcut. The figure shows why reporting every algebraic candidate as a solution without checking the original equation leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is reporting every algebraic candidate as a solution without checking the original equation.

Check yourself

Solve $x^2-7x+12=0$

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice

Ten concrete questions

Practice 101

Solve $x^2-7x+12=0$

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 202

Solve $|2x-1|=7$

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 303

Solve $3^(x+1)=81$

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 404

Solve $4-2x\le 10$

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 505

Explain why this conclusion is valid: The domain requires $x\ge 1$ . Squaring gives candidates $0$ and $3$ ; only $x=3$ checks. Use the foundation problem as evidence: Solve $sqrt(x+1)=x-1$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 606

Solve the representation example, then name the feature of equation and inequality repair that it illustrates: Solve $|2x-1|=7$

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is reporting every algebraic candidate as a solution without checking the original equation.

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Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 808

Connect two representations for this example: Solve $sqrt(x+1)=x-1$ . Describe what a graph, table, mapping, or algebraic form would have to show.

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Practice 909

Create a nearby example by changing one number or condition in this prompt: Solve $3^(x+1)=81$ . Predict the effect, solve your new example, and compare it with the original.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 1010

Write a short verification checklist for equation and inequality repair, then apply it to one worked example from this lesson.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Lesson close

Connect forward

The next lesson, Factoring and polynomial structure repair, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Stitz and Zeager, Precalculus Chapter 0
BetterGrades Algebra course
Redden, Advanced Algebra

No long source passage is reproduced.

Solvesqrt(x+1)=x−1sqrt(x+1)=x-1sqrt(x+1)=x−1

Solvesqrt(x+1)=x−1sqrt(x+1)=x-1sqrt(x+1)=x−1

Solve∣2x−1∣=7|2x-1|=7∣2x−1∣=7

Solve3(x+1)=813^(x+1)=813(x+1)=81

Original BetterGrades manuscript, rights-separated references.

Solve $sqrt(x+1)=x-1$

Solve $sqrt(x+1)=x-1$

Solve $|2x-1|=7$

Solve $3^(x+1)=81$