Example Question - absolute value equations
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Solving Absolute Value Inequalities
To solve the inequality \( 8 - 7|1 - 6s| < -6 \), follow these steps: 1. First, isolate the absolute value term on one side of the inequality: \( 8 - 7|1 - 6s| + 6 < 0 \) \( 14 - 7|1 - 6s| < 0 \) \( -7|1 - 6s| < -14 \) Now divide by -7, remembering that dividing by a negative number will flip the inequality sign: \( |1 - 6s| > 2 \) 2. Next, we need to consider the two cases due to the absolute value, where \(1 - 6s\) can be either greater than 2 or less than -2. **Case 1 - Positive case:** \( 1 - 6s > 2 \) Subtract 1 from both sides: \( -6s > 1 \) Divide by -6 and flip the inequality sign: \( s < \frac{1}{6} \) **Case 2 - Negative case:** \( 1 - 6s < -2 \) Subtract 1 from both sides: \( -6s < -3 \) Divide by -6 and flip the inequality sign: \( s > \frac{1}{2} \) 3. Combine both cases to get the final solution: \( s < \frac{1}{6} \) or \( s > \frac{1}{2} \) The solution to the inequality is all \( s \) that satisfy either \( s < \frac{1}{6} \) or \( s > \frac{1}{2} \).
Solving Absolute Value Equations
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) = 28 Now, let's solve for x: 8x + 32 = 28 Subtract 32 from both sides: 8x = 28 - 32 8x = -4 Divide both sides by 8: x = -4/8 x = -0.5 2. 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 = 28 Add 32 to both sides: -8x = 28 + 32 -8x = 60 Divide both sides by -8: x = 60 / (-8) x = -7.5 So 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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