Select all of the inequalities below where -6 is a solution.
-x<7
x+2 \geq-3
-5 \leq x-3
3 x-1<17
x+4 \geq-7
4 x \geq 13
Answer (1)
Substitute -6 into each inequality.
Check if the resulting statement is true.
The inequalities where -6 is a solution are: − x < 7 , − 5 ≤ x − 3 , 3 x − 1 < 17 , and x + 4 ≥ − 7 .
The solution is − x < 7 , − 5 ≤ x − 3 , 3 x − 1 < 17 , x + 4 ≥ − 7 .
Explanation
Problem Analysis We need to check each inequality to see if -6 is a solution. We will substitute -6 for x in each inequality and determine if the resulting statement is true.
Checking the first inequality
− x < 7 : Substitute x = − 6 to get − ( − 6 ) < 7 , which simplifies to 6 < 7 . This is true, so -6 is a solution.
Checking the second inequality
= -3"> x + 2" >= − 3 : Substitute x = − 6 to get = -3"> − 6 + 2" >= − 3 , which simplifies to = -3"> − 4" >= − 3 . This is false, so -6 is not a solution.
Checking the third inequality
= x-3"> − 5" >= x − 3 : Substitute x = − 6 to get = -6 - 3"> − 5" >= − 6 − 3 , which simplifies to = -9"> − 5" >= − 9 . This is true, so -6 is a solution.
Checking the fourth inequality
3 x − 1 < 17 : Substitute x = − 6 to get 3 ( − 6 ) − 1 < 17 , which simplifies to − 18 − 1 < 17 , or − 19 < 17 . This is true, so -6 is a solution.
Checking the fifth inequality
= -7"> x + 4" >= − 7 : Substitute x = − 6 to get = -7"> − 6 + 4" >= − 7 , which simplifies to = -7"> − 2" >= − 7 . This is true, so -6 is a solution.
Checking the sixth inequality
= 13"> 4 x " >= 13 : Substitute x = − 6 to get = 13"> 4 ( − 6 ) " >= 13 , which simplifies to = 13"> − 24" >= 13 . This is false, so -6 is not a solution.
Final Answer The inequalities for which -6 is a solution are: − x < 7 , − 5 ≤ x − 3 , 3 x − 1 < 17 , and x + 4 ≥ − 7 .
Examples
Understanding inequalities helps in everyday situations like budgeting. For instance, if you want to spend less than 50 o n g rocer i es ( x < 50$), or if you need to save at least 100 e a c hm o n t h ( x >= 100 ) , in e q u a l i t i es h e lp yo u se t an d a c hi e v eyo u r f inan c ia l g o a l s . T h e y ′ re a l so u se f u l in u n d ers t an d in g co n s t r ain t s , l ik e n ee d in g t o b e a tl e a s t 48 in c h es t a llt or i d e a ro ll erco a s t er ( x >= 48$).
