Solve for $(x): |2x-6| < 12$
Answer (2)
Rewrite the absolute value inequality as a compound inequality: − 12 < 2 x − 6 < 12 .
Add 6 to all parts: − 6 < 2 x < 18 .
Divide all parts by 2: − 3 < x < 9 .
Express the solution in interval notation: x in ( − 3 , 9 ) . The solution is x in ( − 3 , 9 ) .
Explanation
Understanding the Problem We are given the absolute value inequality ∣2 x − 6∣ < 12 and we want to solve for x . This means we want to find all values of x that satisfy this inequality.
Rewriting as a Compound Inequality The absolute value inequality ∣2 x − 6∣ < 12 can be rewritten as a compound inequality: − 12 < 2 x − 6 < 12 This is because the absolute value of a number is its distance from zero. So, ∣2 x − 6∣ < 12 means that the distance between 2 x − 6 and 0 is less than 12.
Adding 6 to All Parts To solve the compound inequality, we want to isolate x in the middle. First, we add 6 to all parts of the inequality: − 12 + 6 < 2 x − 6 + 6 < 12 + 6 This simplifies to: − 6 < 2 x < 18
Dividing by 2 Next, we divide all parts of the inequality by 2: f r a c − 6 2 < 2 2 x < 2 18 This simplifies to: − 3 < x < 9
Final Answer The solution to the inequality is all values of x such that − 3 < x < 9 . In interval notation, this is the interval ( − 3 , 9 ) .
Examples
Absolute value inequalities are useful in many real-world scenarios. For example, if you are manufacturing parts for a machine, you might have a tolerance for the size of the parts. Suppose a part is supposed to be 5 cm long, but it can be off by up to 0.1 cm. This means the actual length x must satisfy the inequality ∣ x − 5∣ < 0.1 . Solving this inequality tells you the acceptable range of lengths for the part: 4.9 < x < 5.1 .
To solve the inequality ∣2 x − 6∣ < 12 , we transform it into the compound inequality − 12 < 2 x − 6 < 12 . By isolating x , we find the solution is − 3 < x < 9 or in interval notation, x ∈ ( − 3 , 9 ) .
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