1.2 Linear Equations and Rational Equations

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1.2 Linear Equationsand Rational Equations
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Terms Involving Equations
3x - 1 2 An equation consists of two
algebraic expressions joined by an equal sign.
3x 1 2 3x 3 x 1 1 is a
solution or root of the equation. If you have a
SOLUTION to an equation, that is the value for
the variable(s) that make the equation true.
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Definition of a Linear Equation

• A linear equation in one variable x is an
  equation that can be written in the form
• ax b 0
• where a and b are real numbers and a ? 0.

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Generating Equivalent Equations
An equation can be transformed into an equivalent
equation by one or more of the following
operations.
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Solving a Linear Equation

• OPTIONAL multiply through by LCD or multiple of
  10 to clear fractions or decimals
• SIMPLIFY the algebraic expression on each side.
• ADD to collect all the variable terms on one side
  and all the constant terms on the other side.
• DIVIDE by the coefficient of the variable to
  isolate the variable.
• Check the proposed solution in the original
  equation.

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Ex.
Solve the equation 2(x - 3) - 17 13 - 3(x 2).
Solution
Step 1 SIMPLIFY the algebraic expression on
each side.
Step 2 ADD to collect variable terms on one side
and constant terms on the other side. Step
3 DIVIDE by the coefficient of the variable to
isolate the variable and solve.
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Cont.
Step 4 Check the proposed solution in the
original equation. Substitute 6 for x in the
original equation.
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The solution set is ____.
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Types of Equations

• IdentityAn equation that is true for all real
  numbers. (0 0, all real numbers)
• Conditional An equation that is true for at
  least one real number. (x 0, or any constant)
• Inconsistent An equation that is not true for
  any real number. (0 5, NO SOLUTION)

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Example
Determine whether the equation 3(x - 1) 3x 5
is an identity, a conditional equation, or an
inconsistent equation.
Solution To find out, solve the equation.
3(x 1) 3x 5
This equation is _________________.
Do p 104 27 49 in class.