Solve for $(x) ;|5 x+12|>8$. Graph and interval:
Answer (2)
Split the absolute value inequality into two separate inequalities: 8"> 5 x + 12 > 8 or 5 x + 12 < − 8 .
Solve the first inequality: -4"> 5 x > − 4 , which gives -\frac{4}{5}"> x > − 5 4 .
Solve the second inequality: 5 x < − 20 , which gives x < − 4 .
Express the solution in interval notation: ( − ∞ , − 4 ) ∪ ( − 5 4 , ∞ ) .
Explanation
Problem Analysis We are asked to solve the absolute value inequality 8"> ∣5 x + 12∣ > 8 , graph the solution on a number line, and express the solution in interval notation.
Splitting into Cases The absolute value inequality 8"> ∣5 x + 12∣ > 8 is equivalent to two separate inequalities:
8"> 5 x + 12 > 8 or 5 x + 12 < − 8 .
Solving the First Inequality Let's solve the first inequality, 8"> 5 x + 12 > 8 . Subtracting 12 from both sides gives:
8 - 12"> 5 x > 8 − 12 , which simplifies to -4"> 5 x > − 4 . Dividing both sides by 5 gives:
-\frac{4}{5}"> x > − 5 4 .
Solving the Second Inequality Now, let's solve the second inequality, 5 x + 12 < − 8 . Subtracting 12 from both sides gives:
5 x < − 8 − 12 , which simplifies to 5 x < − 20 . Dividing both sides by 5 gives:
x < − 4 .
Combining the Solutions Combining the solutions, we have x < − 4 or -\frac{4}{5}"> x > − 5 4 .
Graphing the Solution The solution set on the number line consists of two open intervals, one extending to the left from -4 and the other extending to the right from -4/5.
Expressing in Interval Notation In interval notation, the solution set is ( − ∞ , − 4 ) ∪ ( − 5 4 , ∞ ) .
Final Answer Therefore, the solution to the inequality 8"> ∣5 x + 12∣ > 8 is x < − 4 or -\frac{4}{5}"> x > − 5 4 , which in interval notation is ( − ∞ , − 4 ) ∪ ( − 5 4 , ∞ ) .
Examples
Absolute value inequalities are useful in various real-world scenarios. For example, in manufacturing, they can be used to set tolerance levels for product dimensions. If a machine is set to produce parts that are 5 cm in length, an absolute value inequality can define the acceptable range of variation, such as ∣ x − 5∣ < 0.1 , where x is the actual length of the part. This ensures that all parts produced are within an acceptable tolerance of 0.1 cm from the target length. Similarly, in finance, absolute value inequalities can be used to model risk. For instance, if an investment is expected to yield a 10% return, an absolute value inequality can define the acceptable deviation from this return, such as ∣ r − 0.10∣ < 0.02 , where r is the actual return. This helps investors understand the potential range of outcomes and manage their risk accordingly.
The solution to the inequality 8"> ∣5 x + 12∣ > 8 is x < − 4 or -\frac{4}{5}"> x > − 5 4 , expressed in interval notation as ( − ∞ , − 4 ) ∪ ( − 5 4 , ∞ ) . To solve, we split the absolute value inequality into two separate parts and solve each, obtaining the ranges for x. The solution can also be illustrated on a number line with open circles at -4 and -4/5, shading in the correct directions.
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