Solve the following inequality algebraically. \[ 5|x-2|-7 \leq 43 \]
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To solve the inequality \( 5|x-2|-7 \leq 43 \), we first isolate the absolute value term. Add 7 to both sides: \[ 5|x-2| \leq 50 \] Next, divide both sides by 5: \[ |x-2| \leq 10 \] This gives us two cases to consider: 1. \( x - 2 \leq 10 \) 2. \( x - 2 \geq -10 \) Solving the first case: \[ x - 2 \leq 10 \implies x \leq 12 \] Solving the second case: \[ x - 2 \geq -10 \implies x \geq -8 \] Combining both results, we have: \[ -8 \leq x \leq 12 \] Thus, the solution to the inequality \( 5|x-2|-7 \leq 43 \) is: \[ [-8, 12] \]