Solve: $\left|\frac{x}{6}-\frac{7}{12}\right|=\frac{11}{6}$ Negative Direction: $\frac{x}{6}-\frac{7}{12}=-\frac{11}{6}$ $\frac{x}{6}=-\frac{15}{12}$ Determine the values for $x$ by completing the steps shown. Positive Direction: $\frac{x}{6}-\frac{7}{12}=\frac{11}{6}$ $\frac{x}{6}=\frac{29}{12}$ $x=\frac{29}{2}$ or $x=-\frac{15}{2}$ $x=-\frac{29}{2}$ or $x=\frac{15}{2}$ $x=-\frac{29}{2}$ or $x=-\frac{15}{2}$ $x=\frac{39}{2}$ or $x=\frac{15}{2}$
Answer (2)
Split the absolute value equation into two cases: 6 x − 12 7 = 6 11 and 6 x − 12 7 = − 6 11 .
Solve the first case for x : 6 x = 12 29 , which gives x = 2 29 .
Solve the second case for x : 6 x = − 12 15 , which gives x = − 2 15 .
The solutions are x = 2 29 and x = − 2 15 , so the final answer is x = 2 29 , x = − 2 15 .
Explanation
Understanding the Absolute Value We are given the absolute value equation 6 x − 12 7 = 6 11 . To solve this, we consider two cases: the expression inside the absolute value is either equal to 6 11 or − 6 11 .
Solving the Positive Case Case 1: 6 x − 12 7 = 6 11 .
To solve for x , we first add 12 7 to both sides of the equation: 6 x = 6 11 + 12 7 To add the fractions, we need a common denominator, which is 12. So we rewrite 6 11 as 12 22 :
6 x = 12 22 + 12 7 6 x = 12 29 Now, we multiply both sides by 6 to isolate x :
x = 12 29 × 6 x = 2 29 So, x = 2 29 is one solution.
Solving the Negative Case Case 2: 6 x − 12 7 = − 6 11 .
To solve for x , we first add 12 7 to both sides of the equation: 6 x = − 6 11 + 12 7 To add the fractions, we need a common denominator, which is 12. So we rewrite − 6 11 as − 12 22 :
6 x = − 12 22 + 12 7 6 x = − 12 15 Now, we multiply both sides by 6 to isolate x :
x = − 12 15 × 6 x = − 2 × 2 15 × 2 × 3 x = − 2 15 So, x = − 2 15 is another solution.
Final Answer Therefore, the solutions to the equation 6 x − 12 7 = 6 11 are x = 2 29 and x = − 2 15 .
Examples
Absolute value equations are useful in many real-world scenarios, such as calculating tolerances in engineering or determining distances in physics. For example, if you are manufacturing a part that needs to be exactly 5 cm long, but a tolerance of 0.1 cm is acceptable, you can use an absolute value equation to describe the acceptable lengths of the part. The equation would be ∣ x − 5∣ ≤ 0.1 , where x is the length of the part. Solving this equation would give you the range of acceptable lengths: 4.9 ≤ x ≤ 5.1 .
The equation 6 x − 12 7 = 6 11 is solved by considering two cases: one where the expression is equal to 6 11 and another where it equals − 6 11 . The solutions found are x = 2 29 and x = − 2 15 .
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