Question

How do you solve the inequality \[3x + 4 < 31\]?

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 ‘x’ 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 \[3x + 4 < 31\]
We need to solve for ‘x’.
Since we know that the direction of inequality doesn’t change if we subtract a number on both sides. We subtract 4 on both sides of the inequality we have,
\[3x + 4 - 4 < 31 - 4\]
\[3x < 27\]
Divide by 3 on both side of the inequality,
\[x < \dfrac{{27}}{3}\]
\[x < 9\]
We can write it in the interval form. That is \[( - \infty ,9)\].

Note: If we take a value of ‘x’ in \[( - \infty ,9)\] and put it in \[3x + 4 < 31\], it satisfies. That is
Let put \[x = 1\] in \[3x + 4 < 31\],
\[3(1) + 4 < 31\]
\[7 < 31\]
That is 7 is less than 31 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’.

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.