Question

Solve the inequality: $2x - 5 \leqslant 5x + 4 < 11$ where $x \in 1$. Also represent the solution set on the number line.

Answer 1

In the question, we are given an inequation. A statement involving variable(s) and the sign of inequalities is called an inequation or inequality. As we can see the given inequation can be separated into two inequations or it can be solved simultaneously. Here, we are solving them separately because calculations become easier while solving inequalities separately. We will find the value of $x$ from both of the inequalities and find the common interval between them. The common interval will be the solution set of $2x - 5 \leqslant 5x + 4 < 11$ inequation.

Complete step-by-step solution:
We have,
$2x - 5 \leqslant 5x + 4 < 11$
$\Rightarrow 2x - 5 \leqslant 5x + 4$ and $5x + 4 < 11$
Thus, we have two inequalities. Now, we will solve them separately.
Inequation: $2x - 5 \leqslant 5x + 4$
Transport $2x$ to RHS and $4$ to LHS
$- 5 - 4 \leqslant 5x - 2x$
$- 9 \leqslant 3x$
Divide both sides by $3$
$\dfrac{{ - 9}}{3} \leqslant \dfrac{{3x}}{3}$
$- 3 \leqslant x$
It can also be written as:
$x \geqslant - 3$
So, the solution set of inequation $2x - 5 \leqslant 5x + 4$ is $\left[ { - 3,\infty } \right)$
Inequation: $5x + 4 < 11$
Transport $4$ to RHS
$\Rightarrow 5x < 11 - 4$
$\Rightarrow 5x < 7$
Divide both sides by $5$
$\Rightarrow \dfrac{{5x}}{5} < \dfrac{7}{5}$
$\Rightarrow x < \dfrac{7}{5}$
$\Rightarrow x < 1.4$
So, the solution set of inequation $5x + 4 < 11$ is the interval $\left( { - \infty ,1.4} \right)$
Since $x \in 1$, the solution set for inequation $2x - 5 \leqslant 5x + 4 < 11$ is \left\\{ { - 3, - 2, - 1,0,1} \right\\}
The solution set can be represented on the number line as follows:

Note: Addition or subtraction of the same number to both sides of an inequation doesn’t affect the sign of inequality. Both sides of an inequation can be multiplied or divided by the same positive real number without changing the sign of inequality but the sign of inequality is reversed when both sides of an inequation are multiplied or divided by a negative number. Any term of an inequation can be taken to the other side with its sign changed without affecting the sign of inequality.