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|3x+4| > 6-7x.

$|3 x+4|>6-7 x$ .

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Integer inequalities with absolute values

Full solution

Q.

|3 x+4|>6-7 x

.

Consider Cases: First, we need to consider the two cases for the absolute value inequality. $\newline$ Case $1$ : $3x + 4 > 6 - 7x$ when $(3x + 4)$ is positive.
Addition and Isolation: Add $7x$ to both sides to get all the $x$ terms on one side. $\newline$ $3x + 4 + 7x > 6 - 7x + 7x$ $\newline$ $10x + 4 > 6$
Division and Solution: Subtract $4$ from both sides to isolate the term with $x$ . $\newline$ $10x + 4 - 4 > 6 - 4$ $\newline$ $10x > 2$
Consider Negative Case: Divide both sides by $10$ to solve for $x$ . $\frac{10x}{10} > \frac{2}{10}$ $x > \frac{1}{5}$
Multiplication and Inequality: Now, let's consider Case $2$ : $-(3x + 4) > 6 - 7x$ when $(3x + 4)$ is negative.
Subtraction and Isolation: Multiply both sides by $-1$ and remember to flip the inequality sign. $\newline$ $-1 \times -(3x + 4) < -1 \times (6 - 7x)$ $\newline$ $3x + 4 < 7x - 6$
Addition and Isolation: Subtract $3x$ from both sides to get all the $x$ terms on one side. $\newline$ $3x + 4 - 3x < 7x - 6 - 3x$ $\newline$ $4 < 4x - 6$
Division and Solution: Add $6$ to both sides to isolate the term with $x$ . $\newline$ $4 + 6 < 4x - 6 + 6$ $\newline$ $10 < 4x$
Combine Solutions: Divide both sides by $4$ to solve for $x$ . $\newline$ $\frac{10}{4} < \frac{4x}{4}$ $\newline$ $2.5 < x$
Combine Solutions: Divide both sides by $4$ to solve for $x$ . $\newline$ $\frac{10}{4} < \frac{4x}{4}$ $\newline$ $2.5 < x$ Combine the solutions from both cases to get the final solution for the inequality. $\newline$ $x > \frac{1}{5}$ or $x > 2.5$

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