Question

How do you solve the inequality \[\dfrac{1}{2}a - 1 < 3\]?

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 ‘a’ 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 \[\dfrac{1}{2}a - 1 < 3\]
We need to solve for ‘a’.
Since we know that the direction of inequality doesn’t change if we add a number on both sides. We add 1 on both sides of the inequality we have,
\[\dfrac{1}{2}a - 1 + 1 < 3 + 1\]
\[\dfrac{1}{2}a < 4\]
Multiply 2 on both side of the inequality we have
\[\dfrac{1}{2} \times 2a < 4 \times 2\]
\[ \Rightarrow a < 8\]
Thus the solution of \[\dfrac{1}{2}a - 1 < 3\] is \[a < 8\].
We can write it in the interval form. That is \[( - \infty ,8)\].

Note: If we take a value of ‘a’ in \[( - \infty ,8)\] and put it in \[\dfrac{1}{2}a - 1 < 3\], it satisfies. That is
Let put \[a = 1\] in \[\dfrac{1}{2}a - 1 < 3\],
\[\Rightarrow \dfrac{1}{2}(1) - 1 < 3\]
\[\Rightarrow \dfrac{1}{2} - 1 < 3\]
\[\Rightarrow \dfrac{{1 - 2}}{2} < 3\]
\[ \Rightarrow - \dfrac{1}{2} < 3\]
\[ \Rightarrow - 0.5 < 3\]
That is -0.5 is less than 3 and it is correct.
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’.

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.