The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars: If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84. $∣ 2 x^{2} - 4 ∣ = ∣ 2 x^{2} ∣$

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Recognize that the equation |2x^2 - 4| = |2x^2| fits the form |u| = |v|, where u = 2x^2 - 4 and v = 2x^2.

Apply the property that |u| = |v| implies u = v or u = -v. This gives two separate equations to solve:

\[2x^2 - 4 = 2x^2\] and \[2x^2 - 4 = -2x^2\].

Solve the first equation \[2x^2 - 4 = 2x^2\] by isolating terms and simplifying to find possible values of x.

Solve the second equation \[2x^2 - 4 = -2x^2\] by moving all terms to one side to form a quadratic equation, then solve for x.

Check all solutions in the original equation to ensure they satisfy the absolute value equality.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Absolute Value Definition and Properties

Absolute value represents the distance of a number from zero on the number line, always non-negative. For any expression u, |u| = u if u ≥ 0, and |u| = -u if u < 0. Understanding this helps rewrite equations involving absolute values into equivalent forms without the bars.

Equations Involving Two Absolute Values

When an equation has two absolute values set equal, such as |u| = |v|, it can be rewritten as two separate equations: u = v or u = -v. This property allows solving complex absolute value equations by breaking them into simpler cases.

Solving Quadratic Equations

The expressions inside the absolute values may be quadratic, requiring techniques like factoring, using the quadratic formula, or simplifying to find solutions. Recognizing and solving these quadratic equations is essential to find all possible values of the variable.