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. |4x - 3| = |4x - 5|

검증된 단계별 안내

Recognize that the equation |4x - 3| = |4x - 5| can be rewritten using the property that if |u| = |v|, then u = v or u = -v. Here, let u = 4x - 3 and v = 4x - 5.

Set up the two separate equations based on the property: 4x - 3 = 4x - 5 and 4x - 3 = -(4x - 5).

Solve the first equation: 4x - 3 = 4x - 5. Subtract 4x from both sides to isolate constants and check for solutions.

Solve the second equation: 4x - 3 = -4x + 5. Combine like terms by adding 4x to both sides and adding 3 to both sides to isolate x.

Check the solutions obtained from both equations to ensure they satisfy the original absolute value equation.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Absolute Value Definition

The absolute value of a number or expression represents its distance from zero on the number line, always yielding a non-negative result. For any expression u, |u| equals u if u is non-negative, and -u if u is negative. This concept is fundamental when dealing with equations involving absolute values.

Equations Involving Two Absolute Values

When an equation has two absolute value expressions set equal, such as |u| = |v|, it implies that either u = v or u = -v. This property allows us to rewrite the equation without absolute value bars and solve two separate linear equations, covering all possible solutions.

Solving Linear Equations

After rewriting the absolute value equation as two linear equations, solving each involves isolating the variable using algebraic operations like addition, subtraction, multiplication, or division. Understanding how to manipulate and solve linear equations is essential to find all valid solutions.