Which number line is correct for the inequality $-2
\leq (1 / x)$
Answer (2)
Solve the inequality − 2 ≤ ( 1/ x ) by considering cases 0"> x > 0 and x < 0 .
For 0"> x > 0 , the inequality always holds.
For x < 0 , the inequality becomes x ≤ − 1/2 .
Combine the solutions: x ≤ − 1/2 or 0"> x > 0 . The final answer is x ≤ − 2 1 or 0"> x > 0 .
Explanation
Problem Analysis We are given the inequality − 2 ≤ ( 1/ x ) and asked to determine the correct number line representation of its solution.
Considering Cases To solve the inequality, we consider two cases: 0"> x > 0 and x < 0 .
Case 1: x > 0 Case 1: 0"> x > 0 . In this case, 1/ x is always positive. Since − 2 is negative, the inequality − 2 ≤ ( 1/ x ) is always true for any positive x . Thus, all 0"> x > 0 are part of the solution.
Case 2: x < 0 Case 2: x < 0 . In this case, 1/ x is negative. To solve the inequality, we can multiply both sides by x . Since x < 0 , we must reverse the inequality sign: − 2 x ≥ 1 . Dividing both sides by − 2 (and again reversing the inequality sign) gives x ≤ − 1/2 .
Combining the Cases Combining the two cases, the solution to the inequality is x ≤ − 1/2 or 0"> x > 0 . This means that x can be any number less than or equal to − 1/2 , or any number greater than 0 . Note that x cannot be equal to 0 because 1/ x would be undefined.
Number Line Representation The number line should show a closed interval from − ∞ to − 1/2 (inclusive) and an open interval from 0 (exclusive) to ∞ . The given number line shows an interval from -5 to 5. The solution x ≤ − 1/2 is represented by the interval from -5 to -1/2, including -1/2. The solution 0"> x > 0 is represented by the interval from 0 to 5, excluding 0.
Final Answer The number line should have a closed circle at − 1/2 = − 0.5 and an open circle at 0 . The line should be shaded to the left of − 0.5 and to the right of 0 .
Examples
Understanding inequalities like − 2 ≤ ( 1/ x ) is crucial in many real-world applications, such as determining the stability of electrical circuits or analyzing the behavior of physical systems. For instance, in circuit analysis, the current flowing through a resistor might be inversely proportional to the resistance. If we need to ensure the current stays within a certain bound, we can use inequalities to determine the acceptable range of resistance values. Similarly, in physics, analyzing the motion of objects often involves inequalities to define constraints on velocity or acceleration, ensuring the system remains stable and predictable. These mathematical tools help engineers and scientists design and control systems effectively.
The inequality − 2 ≤ x 1 is solved by considering two cases: for 0"> x > 0 , the inequality is always true, and for x < 0 , it simplifies to x ≤ − 2 1 . The solution is then x ≤ − 2 1 or 0"> x > 0 . The number line includes a closed dot at − 2 1 and an open dot at 0 with appropriate shadings.
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