Example 5 - Cont.
[tex]j^2\ \textgreater \ 25 \quad \rightarrow \quad j^2-25\ \textgreater \ 0[/tex]
Step 3: Mark the boundary numbers: [blank] and [blank].
Step 4: Check the signs (+ or -) for a test value in each interval.
Answer (1)
Factor the inequality: 0"> j 2 − 25 = ( j − 5 ) ( j + 5 ) > 0 .
Find the boundary numbers: j = − 5 and j = 5 .
Check the sign of ( j − 5 ) ( j + 5 ) in the intervals ( − ∞ , − 5 ) , ( − 5 , 5 ) , and ( 5 , ∞ ) .
The boundary numbers are − 5 and 5 .
Explanation
Understanding the Problem We are given the inequality 25"> j 2 > 25 , which is rewritten as 0"> j 2 − 25 > 0 . Our goal is to find the boundary numbers and describe the next step in solving this inequality.
Factoring the Inequality To find the boundary numbers, we first factor the left side of the inequality: j 2 − 25 = ( j − 5 ) ( j + 5 ) .
Finding the Boundary Numbers Next, we set each factor to zero to find the boundary numbers: j − 5 = 0 and j + 5 = 0 . Solving these equations gives us j = 5 and j = − 5 . Therefore, the boundary numbers are -5 and 5.
Checking the Signs in Each Interval Now we need to check the sign of the expression ( j − 5 ) ( j + 5 ) in each of the intervals ( − ∞ , − 5 ) , ( − 5 , 5 ) , and ( 5 , ∞ ) . We do this by choosing a test value in each interval and substituting it into the expression. For example, we can choose -6, 0, and 6 as test values.
Conclusion In summary, the boundary numbers are -5 and 5. The next step is to check the sign of the expression ( j − 5 ) ( j + 5 ) in the intervals ( − ∞ , − 5 ) , ( − 5 , 5 ) , and ( 5 , ∞ ) by choosing a test value in each interval.
Examples
Understanding inequalities is crucial in many real-world scenarios, such as determining the range of acceptable values in engineering designs or financial analyses. For example, an engineer might use inequalities to ensure that a structure can withstand a certain range of loads, or a financial analyst might use inequalities to determine the range of investment returns that would meet a certain financial goal. In this case, solving the inequality 25"> j 2 > 25 helps us find the values of j that satisfy the condition, which could represent a physical constraint or a performance requirement.
