Linear inequalities
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In this section ask-math teaches you how to solve linear inequalities with one variable.Some times its not possible to translate word problems in the form of equation so in such a case we may get certain statement involving a sign of inequality which are called 'inequations'
If x and y are any two quantities, in the equations there is x = y but if we get x ≠ y then it is called inequality.
If x ≠ y then any one of the following four conditions (relations) will be true.
either (i) x > y (ii) x$\geq$ y (iii) x < y or (iv) x $\leq$ y
Each of the above four conditions is an inequation.
In the same way, each of the following also represents an ineqation :
x < 8 ; x $\geq$ 6 ; x + 3 $\leq$7 7 ; -x + 2 $\leq$ 11 etc.
Linear inequalities in one variable
If a,b and c are real numbers, then each of the following is called a linear inequation in one variable : (i) ax - b > c , Read as ax - b is greater than c.(ii) ax + b < c, Read as ax + b is less than c.
(iii) ax + b $\geq$ , Read as ax + b is greater than or equal c.
(iv) ax + b $\leq$ , Read as ax + b is less than or equal c.
In linear inequalities, the signs '>', '<', '$\leq$' and $\geq$ are called signs of inequality.
Statement | Meaning of statement | Statement in mathematical form |
x is greater than a | x is any value which is more than 'a' but not 'a'. | |
x is less than a | x is any value which is less than 'a' but not 'a'. | |
x is at least a | x is any value which is greater than 'a' and 'a' is included in it. | |
x is at most a | x is any value which is less than 'a' and 'a' is included in it. | |
1) The cost of a chair (x) is greater than or equal to $20.
2) x is less than 10.
3) 85 is greater than or equal to x.
4) The value of x is at most 35.
5) The value of x is at least 5.
6) -43 is greater than or equal to x.
7) The sum of 4x and 3x is at least 14.
8) If 3 times a number(x) is increased by 4, the result is at least 22.
9) x is at most 38.
10) Three times p plus 5 is at most 11.
Answers
1) x $\geq$ 20 .
2) x < 10.
3) 85 $\geq$ x .
4) x $\leq$ 35 .
5) x $\geq$ 5 .
6) -43 $\geq$ x .
7) 4x + 3x $\geq$ 14 .
8) 3x + 4 $\geq$ 22 .
9) x $\leq$ 38 .
10) 3p + 5 $\leq$ 11 .
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