Solve for $(x) ; |5x + 10| > 8$. Graph and interval:
Answer (2)
Split the absolute value inequality into two cases: 8"> 5 x + 10 > 8 and 5 x + 10 < − 8 .
Solve the first case: -2"> 5 x > − 2 , which gives -\frac{2}{5}"> x > − 5 2 .
Solve the second case: 5 x < − 18 , which gives x < − 5 18 .
Express the solution in interval notation: ( − ∞ , − 5 18 ) ∪ ( − 5 2 , ∞ ) . The final answer is ( − ∞ , − 5 18 ) ∪ ( − 5 2 , ∞ ) .
Explanation
Understanding the Problem We are asked to solve the absolute value inequality 8"> ∣5 x + 10∣ > 8 , and then express the solution as an interval and a graph. Absolute value inequalities can be a bit tricky, but we'll break it down step by step!
Splitting into Cases The absolute value inequality 8"> ∣5 x + 10∣ > 8 means that the expression 5 x + 10 is either greater than 8 or less than -8. This gives us two separate inequalities to solve.
Solving the First Case Let's solve the first case: 8"> 5 x + 10 > 8 . We want to isolate x , so we first subtract 10 from both sides of the inequality: 8 - 10"> 5 x + 10 − 10 > 8 − 10
-2"> 5 x > − 2
Now, we divide both sides by 5: \frac{-2}{5}"> 5 5 x > 5 − 2
-\frac{2}{5}"> x > − 5 2
Solving the Second Case Now let's solve the second case: 5 x + 10 < − 8 . Again, we subtract 10 from both sides: 5 x + 10 − 10 < − 8 − 10
5 x < − 18
Now, we divide both sides by 5: 5 5 x < 5 − 18
x < − 5 18
Expressing the Solution as an Interval So, our solution is x < − 5 18 or -\frac{2}{5}"> x > − 5 2 . In interval notation, this is ( − ∞ , − 5 18 ) ∪ ( − 5 2 , ∞ ) .
Graphing the Solution To graph this solution, we draw a number line. We mark the points − 5 18 and − 5 2 on the number line. Since the inequalities are strict ( < and "> > , not ≤ or ≥ ), we use open circles at these points. Then, we shade the regions to the left of − 5 18 and to the right of − 5 2 , indicating that all values in these regions are solutions to the inequality.
Final Answer Therefore, the solution to the inequality 8"> ∣5 x + 10∣ > 8 is x < − 5 18 or -\frac{2}{5}"> x > − 5 2 , which in interval notation is ( − ∞ , − 5 18 ) ∪ ( − 5 2 , ∞ ) .
Examples
Absolute value inequalities are useful in many real-world situations. For example, suppose a machine is designed to fill bags with 5 kg of flour, but the actual weight can vary by up to 0.2 kg. This means the weight w of the flour in a bag must satisfy the inequality ∣ w − 5∣ ≤ 0.2 . Solving this inequality tells us the range of acceptable weights for the bags of flour. Similarly, in engineering, absolute value inequalities are used to specify tolerances in manufacturing processes, ensuring that components meet certain specifications within acceptable limits. They also appear in physics when dealing with error bounds in measurements.
To solve 8"> ∣5 x + 10∣ > 8 , we break it into two cases: 8"> 5 x + 10 > 8 leading to -\frac{2}{5}"> x > − 5 2 and 5 x + 10 < − 8 leading to x < − 5 18 . The solution in interval notation is ( − ∞ , − 5 18 ) ∪ ( − 5 2 , ∞ ) . A number line graph would show open circles at these points with shading outside those intervals.
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