How can you solve an equation or inequality in one variable?

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  undo the operations done to the variable

Step-by-step explanation:

"Equation or inequality in one variable" covers a lot of territory. We only know how to solve a few kinds of such equations/inequalities. Most of those methods are studied in Algebra courses.

In general, using the properties of equality and of inequality, we seek to "undo" the operations performed on the variable. This approach generally works for polynomials of low degree (4 or less), and certain log, trig, and exponential equations/inequalities. For trig functions, sometimes substitutions or trig identities are used to transform the equation to a polynomial form whose solution we know how to find.

Equations/inequalities in one variable that mix these kinds of expressions (log and trig, for example, or polynomial and exponential) are generally not solvable using algebraic methods. Modern software can make it practical to find real solutions using graphing and/or iterative techniques. Complex solutions may or may not be found using present technology.

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The full answer to the question can involve years of study. Many books have been written on the subject.

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