Question

How do you solve $3x + 4 \geqslant 5x - 8$ ?

Answer 1

Hint: The given equation is a linear equation. Linear equations are the equations of order one. So first solve the expression ignoring the inequality sign and then write the solution in terms of the inequality. Whenever we negate the equation on both sides of the equation the inequality sign changes or inverses.

Complete step by step solution:
The given expression is $3x + 4 \geqslant 5x - 8$
As we know in algebra, arithmetic operations can only be performed on the coefficients with the same power of $x$ .
Here, in the above question arithmetic operations can be performed on $3x\;$ and $5x\;$ as they have the same power of $x$ which is $1$ .
Similarly, $4, - 8$ are also same degree terms. They are of degree $0$ .
To simplify the equation, let’s start by shifting the coefficients with the same power terms on the same side of the equation.
The equation we are given here is
$3x + 4 \geqslant 5x - 8$
Now, subtracting $4$ on both the sides of the equation
$\Rightarrow 3x + 4 - 4 \geqslant 5x - 8 - 4$
$\Rightarrow 3x \geqslant 5x - 12$
Now, subtracting $5x\;$ from both sides of the equation
$\Rightarrow 3x - 5x \geqslant 5x - 5x - 12$
Here we subtract the same degree terms by writing the variable as it is and then subtracting the constants.
$3x - 5x = x(3 - 5) = - 2x$
On evaluating, we get,
$\Rightarrow - 2x \geqslant - 12$
Dividing both the sides of the equation by $2$ , we get,
$\Rightarrow \dfrac{{ - 2x}}{2} \geqslant \dfrac{{ - 12}}{2}$
$\Rightarrow - x \geqslant - 6$
Now, multiplying with $- 1$ on both sides
In linear algebra, when an equation is multiplied or divided by a negative number the inequality is flipped, that is $\geqslant$ is replaced with $\leqslant$ and $\leqslant$ is replaced by $\geqslant$ .
Following the same, we get,
$\Rightarrow x \leqslant 6$
All the values less than or equal to $6$ satisfy the equation.
The range of $x$ is $(-\infty ,6]$

Hence the value of $x$ satisfying the above inequality is $x \leqslant 6$.

Note: One must be careful about the flipping of inequality when the equation is multiplied or divided by a negative number that is $\geqslant$ is replaced with $\leqslant$ and $\leqslant$ is replaced by $\geqslant$ . The value of $x$ can always be verified by putting it back in the given question.
Now we have the value of $x$ as $x \leqslant 6$
Putting $x = 2$ in the equation since any value of $x$ which is less than or equal to $6$ satisfies the condition.
$\Rightarrow 3(2) + 4 \geqslant 5(2) - 8$
$\Rightarrow 6 + 4 \geqslant 10 - 8$
On further evaluation,
$\Rightarrow 10 \geqslant 2$
This condition is correct because $10\;$ is greater than $2$ .
Hence our solution is correct.