Question

How do you solve the inequality \[12 + 10w \geqslant 8(w + 12)\]?

Answer 1

Hint: An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. Here we need to solve for ‘w’ which is a variable. Solving the given inequality is very like solving equations and we do most of the same thing but we must pay attention to the direction of inequality\[( \leqslant , > )\]. We have a simple linear equation type inequality and we can solve this easily.

Complete step-by-step solution:
Given \[12 + 10w \geqslant 8(w + 12)\]
We need to solve for ‘w’.
Expand the brackets we have,
\[12 + 10w \geqslant 8w + \left( {12 \times 8} \right)\]
\[12 + 10w \geqslant 8w + 96\]
Since we know that the direction of inequality doesn’t change if we add or subtract a number on both sides. We subtract ‘8w’ on both sides of the inequality we have,
\[12 + 10w - 8w \geqslant 8w - 8w + 96\]
\[12 + 2w \geqslant 96\]
Similarly we subtract 12 on both side of the inequality,
\[12 - 12 + 2w \geqslant 96 - 12\]
\[2w \geqslant 84\]
We divide the whole inequality by 2 we have,
\[w \geqslant \dfrac{{84}}{2}\]
\[w \geqslant 42\]
Thus the solution of \[12 + 10w \geqslant 8(w + 12)\] is \[w \geqslant 42\].
We can write it in the interval form. That is \[[42,\infty )\].

Note: If we take a value of ‘w’ in \[[42,\infty )\] and put it in \[12 + 10w \geqslant 8(w + 12)\], it satisfies. That is
Let put \[w = 43\] in \[12 + 10w \geqslant 8(w + 12)\],
\[\Rightarrow 12 + 10(43) \geqslant 8(43 + 12)\]
\[\Rightarrow 12 + 10(43) \geqslant 8(43 + 12)\]
\[\Rightarrow 12 + 430 \geqslant 8(55)\]
\\Rightarrow [442 \geqslant 440\]
That is 442 is greater than 440 and it is correct. We also take \[w = 42\] which will satisfy.

We know that \[a \ne b\] says that ‘a’ is not equal to ‘b’. \[a > b\] means that ‘a’ is less than ‘b’. \[a < b\] means that ‘a’ is greater than ‘b’. These two are known as strict inequality. \[a \geqslant b\] means that ‘a’ is less than or equal to ‘b’. \[a \leqslant b\] means that ‘a’ is greater than or equal to ‘b’.

The direction of inequality do not change in these cases:
i) Add or subtract a number from both sides.
ii) Multiply or divide both sides by a positive number.
iii) Simplify a side.
The direction of the inequality change in these cases:
i) Multiply or divide both sides by a negative number.
ii) Swapping left and right hand sides.