Question

How do you solve\[5x - 7 \geqslant 13\]?

Answer 1

Hint: The given inequation is a linear inequation in one variable. An inequation is different from an equation inasmuch as an inequation compares two expressions holding non-equality rather than holding equality of expressions. Solution of given inequation here may give a range of values for $x$ and not just one single value as result.

Complete step-by-step solution:
The given inequation is \[5x - 7 \geqslant 13\].
We have to find the values of $x$ for which the given inequation holds true.
First we try to simplify the inequation such that the RHS contains no other term than $0$. For this, we subtract $13$ from both sides of the inequation. We get:
\[
   \Rightarrow 5x - 7 - 13 \geqslant 13 - 13 \\
   \Rightarrow 5x - 20 \geqslant 0 \\
\]
Now, to get the value of $x$ we have to get the variable $x$ in its simplest form in the LHS and all other terms in the RHS. Since it is a linear inequation, the simplest form of the variable $x$ would be $x$ itself.
For this we first add $20$ to both the sides,
\[
   \Rightarrow 5x - 20 + 20 \geqslant 0 + 20 \\
   \Rightarrow 5x \geqslant 20 \\
\]
Now we divide both sides by $5$,
\[
   \Rightarrow \dfrac{{5x}}{5} \geqslant \dfrac{{20}}{5} \\
   \Rightarrow x \geqslant 4 \\
\]
Thus, the solution that we get is\[x \geqslant 4\], i.e. for any number greater than or equal to $4$ as the value of $x$ the inequality holds true in the given inequation.

Note: Even in an inequation, adding a number to both sides and subtracting a number from both sides would not disturb the inequation. Also, multiplying or dividing by a positive number both sides of an inequation would not disturb the inequation. However, when multiplying or dividing by a negative number the sign of inequality reverses, i.e. greater than (>) becomes less than (<) and vice-versa. For solution of an inequation we get a range of values as a result.