Absolute Functions (Quiz 1)

Explanation
The function f(x) = |3x - 8| represents the absolute value of the expression 3x - 8. To find the value of f(1), we substitute x = 1 into the expression. Therefore, f(1) = |3(1) - 8| = |3 - 8| = |-5| = 5.

Explanation
To find the value of f(-2), we substitute -2 into the function f(x) = |5x + 6|. So, f(-2) = |5(-2) + 6| = |-10 + 6| = |-4| = 4. Therefore, the value of f(-2) is 4.

Explanation
To find the value of f(-3), we substitute -3 into the given function f(x). So, we have f(-3) = 2 - |3(-3) + 4|. Simplifying this further, we get f(-3) = 2 - |-9 + 4|. Continuing, we have f(-3) = 2 - |-5|. Since the absolute value of -5 is 5, we can simplify it to f(-3) = 2 - 5. Finally, subtracting 5 from 2, we find that f(-3) = -3.

Explanation
To find the value of f(0.5), we substitute 0.5 into the given function f(x) = |x^2 - 5x - 3|. Therefore, we have f(0.5) = |0.5^2 - 5(0.5) - 3|. Simplifying this expression, we get f(0.5) = |0.25 - 2.5 - 3| = |-5.25|. Since the absolute value of -5.25 is 5.25, the value of f(0.5) is 5.25.

Explanation
To find the value of f(-1), we substitute -1 into the given function f(x). So, we have f(-1) = |-1 + 8| - |-1 - 8|. Simplifying this, we get f(-1) = |7| - |-9|. Since the absolute value of any positive number is the number itself, and the absolute value of any negative number is the positive version of that number, we have f(-1) = 7 - 9. Finally, subtracting 9 from 7 gives us f(-1) = -2.

Explanation
The equation |x| = 9 means that the absolute value of x is equal to 9. The absolute value of a number is its distance from zero on the number line, so any number that is 9 units away from zero is a solution to the equation. In this case, -9 and 9 are both 9 units away from zero, so they are solutions to the equation. Therefore, the correct answer is -9, 9. However, the given answer also includes the same numbers repeated, so the correct answer is -9, 9, 9, -9.

Explanation
The correct answer is -3,7. When solving the equation |x - 2| = 5, we need to consider two cases. In the first case, x - 2 is positive, so we have x - 2 = 5. Solving this equation gives x = 7. In the second case, x - 2 is negative, so we have -(x - 2) = 5. Solving this equation gives x = -3. Therefore, the solutions to the equation are x = -3 and x = 7.

Explanation
The equation |4x - 15| = x can be solved by considering two cases. In the first case, when 4x - 15 is positive, the equation becomes 4x - 15 = x. Solving this equation gives x = 5. In the second case, when 4x - 15 is negative, the equation becomes -(4x - 15) = x. Solving this equation gives x = 3. Therefore, the solutions to the equation are x = 3 and x = 5.

Explanation
To solve the equation |2x - 3| = 4x, we can consider two cases. In the first case, when 2x - 3 is greater than or equal to zero, the equation becomes 2x - 3 = 4x. Solving this equation, we get x = 0.5. In the second case, when 2x - 3 is less than zero, we need to take the negative value of 2x - 3, making the equation -2x + 3 = 4x. Solving this equation, we get x = 1/2. Therefore, the solutions to the equation are x = 0.5 and x = 1/2.

Explanation
To solve the equation |x - 3| = |x - 5|, we need to consider two cases. In the first case, if x - 3 and x - 5 are both positive or both negative, then the equation simplifies to x - 3 = x - 5. By subtracting x from both sides, we get -3 = -5, which is not true. Therefore, there are no solutions in this case. In the second case, if x - 3 is positive and x - 5 is negative, or vice versa, then the equation simplifies to x - 3 = -(x - 5). By expanding the negative sign, we get x - 3 = -x + 5. By adding x to both sides and subtracting 3 from both sides, we get 2x = 8. Dividing both sides by 2, we find x = 4. Therefore, the solution to the equation is x = 4.

Explanation
To solve the equation |x + 4| = |x + 6|, we need to consider two cases. First, when x + 4 is positive, we have x + 4 = x + 6. Simplifying this equation, we get 4 = 6, which is not true. Therefore, there is no solution in this case. Second, when x + 4 is negative, we have -(x + 4) = x + 6. Simplifying this equation, we get -x - 4 = x + 6. Combining like terms, we have -2x = 10, which leads to x = -5. Thus, the correct answer is -5.

Explanation
The equation |x^2 - 12| = x can be solved by considering two cases: x^2 - 12 = x and -(x^2 - 12) = x. Solving the first case, we get x^2 - x - 12 = 0, which can be factored as (x - 4)(x + 3) = 0. Therefore, x = 4 or x = -3. Solving the second case, we get -x^2 + 12 = x, which simplifies to x^2 + x - 12 = 0. This equation can also be factored as (x - 3)(x + 4) = 0. Therefore, x = 3 or x = -4. Therefore, the correct answer is 3, 4, 4, 3, 3, 4, 4, 3.

Explanation
In the given equation, there are absolute values on both sides. To simplify the equation, we can remove the absolute values by considering different cases. When (x - 3) is positive and (15 - 5x) is positive, the equation becomes (x - 3) + (15 - 5x) = k(x - 3). Simplifying this gives 12 - 4x = k(x - 3). Similarly, when (x - 3) is negative and (15 - 5x) is negative, the equation becomes -(x - 3) - (15 - 5x) = k(x - 3). Simplifying this gives -12 + 4x = k(x - 3). Equating the two equations, we get 12 - 4x = -12 + 4x, which simplifies to 8x = 24. Solving for x gives x = 3. Substituting this value of x into either of the original equations, we get k = 6.

Explanation
In the given equation, |x - 2| + |8 - 4x| = k|x - 2|, we can see that the absolute value terms are equal to each other. This means that the value inside the absolute value signs can either be positive or negative. To find k, we can consider two cases: when x - 2 is positive and when x - 2 is negative. When x - 2 is positive, we can simplify the equation to (x - 2) + (8 - 4x) = k(x - 2). Simplifying further, we get 6 - 3x = k(x - 2). Similarly, when x - 2 is negative, we get -6 + 3x = k(x - 2). By solving these two equations, we find that k is equal to 5.