Solve each equation or inequality.∣4x+3∣\u003e0| 4x+ 3| \u003e 0

Ch. 1 - Equations and Inequalities

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Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 66

2장, 문제 66

Solve each equation or inequality.
$∣ 4 x + 3 ∣ > 0$

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Identify the inequality to solve: $|4x + 3| > 0$.

Recall that the absolute value $|A|$ represents the distance of $A$ from zero on the number line, so $|A| > 0$ means $A$ is not equal to zero.

Set the expression inside the absolute value not equal to zero: $4x + 3 \neq 0$.

Solve the equation $4x + 3 = 0$ to find the value that $x$ cannot be: subtract 3 from both sides to get $4x = -3$, then divide both sides by 4 to get $x = -\frac{3}{4}$.

Conclude that the solution to the inequality $|4x + 3| > 0$ is all real numbers except $x = -\frac{3}{4}$.

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주요 개념

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

Absolute Value Inequalities

Absolute value inequalities involve expressions within absolute value bars and require understanding how to interpret them. For |A| > 0, the expression inside the absolute value, A, must be any number except zero, since absolute value measures distance from zero. This concept helps determine the solution set for inequalities involving absolute values.

Properties of Absolute Value

The absolute value of a number is always non-negative and equals zero only when the number itself is zero. This property means |x| > 0 holds true for all x except x = 0. Recognizing this helps simplify and solve inequalities by focusing on when the inside expression equals zero.

Solving Linear Equations

Solving linear equations involves isolating the variable to find its value. In this problem, setting the inside of the absolute value equal to zero (4x + 3 = 0) identifies critical points that affect the inequality's solution. Understanding how to solve such equations is essential for determining boundary values.

Solve each equation or inequality.∣4x+3∣>0| 4x+ 3| > 0