In algebra when we come across a mathematical sentence something like 3x+4y=0 it is called an equation which has an ‘=’ sign. It is not possible always to equate two values; sometimes the values are bigger or smaller relative to each other.
So there is another type of sentence which is used to show the relative size of two values which is called an inequality. The inequality math sentences use one of the following symbols, ‘>’ greater than, ‘<’ less than, ‘=’ less than or equal to and ‘=’ greater than or equal to. Inequality Examples are x+3<9 5-x="" x="-5;">9 etc.
The key words that are seen in the word problems and their meanings are, ‘At least’ means greater than or equal to; ‘Not more than’ means less than or equal to; ‘More than’ means greater than. Some more Examples of Inequalities are as given below:
The sum of x and 4 is greater than -5
Sum of x and 4 is ‘x+4’, the inequality used here is ‘>’, the resultant is -5
Finally the inequality would be, x+4 > -5
Subtracting 4 on both sides gives, x+4-4> -5 -4; x > -9 would be the final inequality. Here it means x can take all the values that are greater than -9. One point to remember here is -9 is not included as there is no equal to sign in the inequality.
The addition, multiplication and absolute principles of inequalities are as follows:
Inequalities Word Problems
A taxi charges a flat rate of $1.85 in addition to $0.65 per mile. John has not more than $10 to spend on the ride. Write an inequality representing John’s situation and calculate the number of miles John can travel without exceeding his limit
Solution: Here the key words are ‘not more than’ so the inequality would be ‘=’
The variable here is number of miles= m
So, the inequality would be $0.65m + $1.85 = $10
Subtracting 1.85 on both sides, 0.65m+1.85-1.85 = 10-1.85
0.65m = 8.15
Dividing on both sides with 0.65, 0.65m/0.65=8.15/0.65
m = 12.54
John can travel 12 miles without exceeding his limit
So there is another type of sentence which is used to show the relative size of two values which is called an inequality. The inequality math sentences use one of the following symbols, ‘>’ greater than, ‘<’ less than, ‘=’ less than or equal to and ‘=’ greater than or equal to. Inequality Examples are x+3<9 5-x="" x="-5;">9 etc.
The key words that are seen in the word problems and their meanings are, ‘At least’ means greater than or equal to; ‘Not more than’ means less than or equal to; ‘More than’ means greater than. Some more Examples of Inequalities are as given below:
The sum of x and 4 is greater than -5
Sum of x and 4 is ‘x+4’, the inequality used here is ‘>’, the resultant is -5
Finally the inequality would be, x+4 > -5
Subtracting 4 on both sides gives, x+4-4> -5 -4; x > -9 would be the final inequality. Here it means x can take all the values that are greater than -9. One point to remember here is -9 is not included as there is no equal to sign in the inequality.
The addition, multiplication and absolute principles of inequalities are as follows:
- If a>b then a+c>b+c
- If a>b and c is positive, then ac>bc and if c is negative, then ac
- If X is any expression and c any positive integer such that |X|
- If X is any expression and c any positive integer such that |X|>c, this would be same as X>c, X>-c
- If X is any expression and c any positive integer such that |X|
Inequalities Word Problems
A taxi charges a flat rate of $1.85 in addition to $0.65 per mile. John has not more than $10 to spend on the ride. Write an inequality representing John’s situation and calculate the number of miles John can travel without exceeding his limit
Solution: Here the key words are ‘not more than’ so the inequality would be ‘=’
The variable here is number of miles= m
So, the inequality would be $0.65m + $1.85 = $10
Subtracting 1.85 on both sides, 0.65m+1.85-1.85 = 10-1.85
0.65m = 8.15
Dividing on both sides with 0.65, 0.65m/0.65=8.15/0.65
m = 12.54
John can travel 12 miles without exceeding his limit
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