DETERMINING IF A GIVEN VALUE MAKES THE INEQUALITY TRUE
In math, we solve inequalities and give solution in different ways like interval notation and graphing.
To have a clear understanding of the problem, we may have to interpret the part of the solution or sometimes the whole.
Let us see, how solution of an inequality can be interpreted through the following examples.
Example 1 :
Determine if x = 4 can be one of the solutions to the inequality 3x + 5 > 14.
Solution :
Step 1 :
Substitute x = 4 in 3x + 5 > 14
3(4) + 5 > 14 -----> ?
12 + 5 > 14 -----> ?
17 > 14 -----> ?
Step 2 :
Is the above inequality true ?
Yes, Because 17 is greater than 14.
So, x = 4 can be one of the solutions to the inequality 3x + 5 > 14.
Example 2 :
At Gas ‘n’ Wash, gasoline sells for $4.00 a gallon and a car wash costs $12. Harika wants to have her car washed and keep her total purchase under $60. The inequality 4g + 12 < 60 can be used to find g, the number of gallons of gas she can buy. Determine which, if any, of these values is a solution: g = 10; g = 11; g = 12.
Solution :
Case 1 : g = 10
Step 1 :
Substitute g = 10 in 4g + 12 < 60
4(10) + 12 < 60 -----> ?
40 + 12 < 60 -----> ?
50 < 60 -----> ?
Step 2 :
Is the above inequality true ?
Yes, because 50 is less than than 60.
Case 2 : g = 11
Step 1 :
Substitute g = 11 in 4g + 12 < 60
4(11) + 12 < 60 -----> ?
44 + 12 < 60 -----> ?
52 < 60 -----> ?
Step 2 :
Is the above inequality true ?
Yes, because 52 is less than than 60.
Case 2 : g = 12
Step 1 :
Substitute g = 11 in 4g + 12 < 60
4(12) + 12 < 60 -----> ?
48 + 12 < 60 -----> ?
60 < 60 -----> ?
Step 2 :
Is the above inequality true ?
No, because 60 is equal to 60.
Conclusion :
Hence, Harika can buy 10 or 11 gallons of gas but not 12 gallons.
Check :
Solve and graph the inequality.
4g + 12 < 60
Subtract 12 from both sides
(4g + 12) - 12 < (60) - 12
4g < 48
Divide both sides by 4
(4g) / 4 < 48 / 4
g < 12
The closed circle at zero represents the minimum amount she can buy, zero gallons. She cannot buy a negative number of gallons. The open circle at 12 means that she can buy any amount up to but not including 12 gallons.
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