Which of the following inequalities is true when [tex]x=-10[/tex]? Select all that apply.
[tex]9 x-5\ \textgreater \ 84.5[/tex]
[tex]-9 x-5\ \textless \ 84.5[/tex]
[tex]-9 x-5\ \textgreater \ -84.5[/tex]
[tex]-9 x-5\ \textgreater \ 84.5[/tex]
None of the options above are true when [tex]x=-10[/tex].
Answer (2)
Substitute x = − 10 into each inequality.
Evaluate the left-hand side of each inequality.
Compare the left-hand side to the right-hand side to determine if the inequality is true or false.
The inequalities that are true for x = − 10 are -84.5"> − 9 x − 5 > − 84.5 and 84.5"> − 9 x − 5 > 84.5 .
Explanation
Understanding the Problem We are given the value of x = − 10 and we need to determine which of the given inequalities are true when x = − 10 .
Evaluating the Inequalities Let's evaluate each inequality by substituting x = − 10 into each one.
Inequality 1 Inequality 1: 84.5"> 9 x − 5 > 84.5 . Substituting x = − 10 , we get 84.5"> 9 ( − 10 ) − 5 > 84.5 , which simplifies to 84.5"> − 90 − 5 > 84.5 , or 84.5"> − 95 > 84.5 . This is false.
Inequality 2 Inequality 2: − 9 x − 5 < 84.5 . Substituting x = − 10 , we get − 9 ( − 10 ) − 5 < 84.5 , which simplifies to 90 − 5 < 84.5 , or 85 < 84.5 . This is false.
Inequality 3 Inequality 3: -84.5"> − 9 x − 5 > − 84.5 . Substituting x = − 10 , we get -84.5"> − 9 ( − 10 ) − 5 > − 84.5 , which simplifies to -84.5"> 90 − 5 > − 84.5 , or -84.5"> 85 > − 84.5 . This is true.
Inequality 4 Inequality 4: 84.5"> − 9 x − 5 > 84.5 . Substituting x = − 10 , we get 84.5"> − 9 ( − 10 ) − 5 > 84.5 , which simplifies to 84.5"> 90 − 5 > 84.5 , or 84.5"> 85 > 84.5 . This is true.
Conclusion Therefore, the inequalities that are true when x = − 10 are -84.5"> − 9 x − 5 > − 84.5 and 84.5"> − 9 x − 5 > 84.5 .
Examples
Understanding inequalities is crucial in many real-world scenarios. For instance, when budgeting, you might need to ensure that your expenses are less than your income. If your income is represented by I and your expenses by E , the inequality E < I must hold true for a balanced budget. Similarly, in engineering, safety margins are often defined using inequalities to ensure that structures can withstand loads greater than expected. If the expected load is L , and the safety factor is S , the structure must be designed to withstand a load of at least S × L , represented by the inequality S \times L"> L o a d d es i g n > S × L .
The true inequalities when x = − 10 are − 9 x − 5 > − 84.5 and − 9 x − 5 > 84.5 . The inequalities 9 x − 5 > 84.5 and − 9 x − 5 < 84.5 are false. In total, two inequalities are true for x = − 10 .
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