Question - Solving Absolute Value Equations
Solution:
The equation to solve is 8|x + 4| = 28. To solve an equation involving an absolute value, we need to consider the two possible cases that result from the definition of absolute value: one where the expression inside the absolute value is positive, and one where it is negative.Let's write down the two cases:1. x + 4 is positive or zero:In this case, |x + 4| just equals x + 4, so we don't need the absolute value bars. We can write the equation as:8(x + 4) = 28Now, let's solve for x:8x + 32 = 28Subtract 32 from both sides:8x = 28 - 328x = -4Divide both sides by 8:x = -4/8x = -0.52. x + 4 is negative:If x + 4 is negative, then |x + 4| equals -(x + 4). So our equation becomes:8(-(x + 4)) = 28-8x - 32 = 28Add 32 to both sides:-8x = 28 + 32-8x = 60Divide both sides by -8:x = 60 / (-8)x = -7.5So the solutions are x = -0.5 and x = -7.5. Looking at the options provided, the correct answer is:D. x = -7.5 and x = -0.5
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