Solve the following equations with absolute value. ( A ) 3 ∣ 4 x + 1 ∣ + 7 = 2 3\left|4x+1\right|+7=2

Welcome back. For this problem, we'd like to solve the equation three times the absolute value of four x plus one plus seven equals two. And we know that the first step to solving an absolute value equation is to isolate the absolute value, so let's start by working to get this absolute value by itself on the left side. The first thing we should do is subtract seven on both sides of the equation. To get rid of this plus seven on the left, leaving us with three times the absolute value of four x plus one, On the right side of the equation, we get two minus seven, which is negative five. Now, we have to get rid of this three, which is being multiplied by the absolute value, so we can get rid of it by dividing both sides of the equation by three. This will cancel with our three on the left, leaving us with absolute value of four x plus one is equal to negative five thirds. Negative five thirds is negative, so we get an equation of the form absolute value equals negative number, which we know by our special cases rule means that there is no solution to this equation. This is because absolute value can never be negative, so no matter what we plug in for x, we're not gonna get a true statement. So, again, this equation has no solution. Great work.

Solve the following equations with absolute value. ( A ) 3 ∣ 4 x + 1 ∣ + 7 = 2 3\left|4x+1\right|+7=2

x = { 1 2 , − 1 } x=\left\lbrace\frac12,-1\right\rbrace

x = { − 1 6 , 2 3 } x=\left\lbrace-\frac16,\frac23\right\rbrace

x = { 1 6 , − 2 3 } x=\left\lbrace\frac16,-\frac23\right\rbrace

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9. Inequalities and Absolute Value

Absolute Value Equations

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Multiple Choice

Solve the following equations with absolute value.
(A) $3 ∣ 4 x + 1 ∣ + 7 = 2$

$x = {\frac{1}{2}, - 1}$

$x = {- \frac{1}{6}, \frac{2}{3}}$

$x = {\frac{1}{6}, - \frac{2}{3}}$

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Verified step by step guidance

Start with the given equation: \$3\left|4x + 1\right| + 7 = 2$.

Isolate the absolute value expression by subtracting 7 from both sides: \$3\left|4x + 1\right| = 2 - 7$.

Simplify the right side: \$3\left|4x + 1\right| = -5$.

Divide both sides by 3 to solve for the absolute value: $\left|4x + 1\right| = \frac{-5}{3}$.

Recognize that the absolute value of any expression is always non-negative, so it cannot equal a negative number. Therefore, there is no solution to this equation.

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Solve the following equations with absolute value.(A) 3∣4x+1∣+7=23\left|4x+1\right|+7=2

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Solve the following equations with absolute value.
(A) $3 ∣ 4 x + 1 ∣ + 7 = 2$