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How do you solve this equation $2x-1>4-\dfrac{1}{2}x$ ?

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Last updated date: 01st Mar 2024
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IVSAT 2024
Answer
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Hint: To solve the given inequality, we bring the $x$ terms to the left side and the arithmetic terms to the right hand side. After doing so, we simplify the inequality by adding, subtracting and so on as required and then, we reach the solution.

Complete step by step answer:
The given inequality is
 $2x-1>4-\dfrac{1}{2}x$
We have to solve this inequality simply just as we solve an equation. For this, we need to bring all the $x$ related terms to one side and other arithmetic terms to the other side. So, at first, we add $\dfrac{1}{2}x$ to both sides of the above inequality. The inequality thus becomes,
$\Rightarrow 2x+\dfrac{1}{2}x-1>4-\dfrac{1}{2}x+\dfrac{1}{2}x$
This upon simplification gives,
$\Rightarrow 2x+\dfrac{1}{2}x-1>4$
Let us multiply both sides of the above inequality by $2$ . Doing so, the above inequality thus becomes,
$\Rightarrow 2\times \left( 2x+\dfrac{1}{2}x-1 \right)>4\times 2$
This upon simplification gives
$\Rightarrow 4x+2\times \dfrac{1}{2}x-2>8$
Further simplifying the above inequality, the above inequality thus becomes,
$\Rightarrow 4x+x-2>8$
Adding the $4x$ and the $x$ terms, the above inequality thus becomes,
$\Rightarrow 5x-2>8$
Adding $2$ to both sides of the above inequality, the above inequality thus becomes,
$\Rightarrow 5x-2+2>8+2$
Simplifying the above inequality, the above inequality thus becomes,
$\Rightarrow 5x>10$
Dividing both sides of the above inequality by $5$ , the above inequality thus becomes,
$\Rightarrow \dfrac{5x}{5}>\dfrac{10}{5}$
Simplifying the above inequality, the above inequality thus becomes,
$\Rightarrow x>2$

Therefore, we can conclude that the solution for the given inequality $2x-1>4-\dfrac{1}{2}x$ is $x>2$.

Note: We should be careful while bringing the $x$ terms to one side and while multiplying or dividing the two sides by a negative number or during taking reciprocals of both sides, as while doing so, we should reverse the inequality sign. We can also solve this problem by taking the two sides of the inequality as two equations of a line. We plot the two lines. Now, the value of $x$ for which the line $y=2x-1$ lies above the line $y=4-\dfrac{1}{2}x$ , is the solution of the inequality. This occurs for all values of $x>2$ , so it’s a solution.
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