Question

How do you solve $\dfrac{{x - 1}}{{2x + 3}} \leqslant 1$?

Answer 1

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 $\dfrac{{x - 1}}{{2x + 3}} \leqslant 1$
We need to solve for ‘x’.
Since we know that the direction of inequality doesn’t change if we multiply the same positive number on both sides. We multiply $2x + 3$ on both sides, we have,
$\Rightarrow x - 1 \leqslant 2x + 3$
Similarly we subtract 3 on both side of the inequality,
$\Rightarrow x - 3 - 1 \leqslant 2x$
Similarly we subtract 3 on both side of the inequality,
$\Rightarrow - 3 - 1 \leqslant 2x - x$
$\Rightarrow - 4 \leqslant x$
$\Rightarrow x \geqslant - 4$
Thus the solution of $\dfrac{{x - 1}}{{2x + 3}} \leqslant 1$ is $\Rightarrow x \geqslant - 4$.
We can write it in the interval form. That is $[ - 4,\infty )$.

Note: If we take a value of ‘w’ in $[ - 4,\infty )$ and put it in $\dfrac{{x - 1}}{{2x + 3}} \leqslant 1$, it satisfies. That is
Let put $x = - 4$ in $\dfrac{{x - 1}}{{2x + 3}} \leqslant 1$,
$\dfrac{{ - 4 - 1}}{{2\left( { - 4} \right) + 3}} \leqslant 1$
$\dfrac{{ - 5}}{{ - 8 + 3}} \leqslant 1$
$\dfrac{{ - 5}}{{ - 5}} \leqslant 1$
$1 \leqslant 1$, which is true. Hence it satisfies .
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:
- Add or subtract a number from both sides.
- Multiply or divide both sides by a positive number.
- Simplify a side.
- The direction of the inequality change in these cases:
- Multiply or divide both sides by a negative number.
- Swapping left and right hand sides.