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$|4x-6| +9>11$

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Integer inequalities with absolute values

Full solution

Q.

|4x-6| +9>11

Isolate absolute value: First, let's isolate the absolute value by subtracting $9$ from both sides of the inequality. $\newline$ $|4x - 6| + 9 - 9 > 11 - 9$ $\newline$ $|4x - 6| > 2$
Consider two cases: Now we need to consider the two cases for the absolute value. $\newline$ Case $1$ : $4x - 6 > 2$ $\newline$ Case $2$ : $-(4x - 6) > 2$
Solve Case $1$ : Let's solve Case $1$ : $\newline$ $4x - 6 > 2$ $\newline$ Add $6$ to both sides: $\newline$ $4x > 8$ $\newline$ Now divide by $4$ : $\newline$ $x > 2$
Solve Case $2$ : Now let's solve Case $2$ : $\newline$ $- (4x - 6) > 2$ $\newline$ First, distribute the negative sign: $\newline$ $-4x + 6 > 2$ $\newline$ Now subtract $6$ from both sides: $\newline$ $-4x > -4$ $\newline$ Divide by $-4$ , remembering to flip the inequality sign: $\newline$ $x < 1$

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