Question

How do you solve $ - 7x \geqslant 56$?

Answer 1

Hint: These questions can be solved by taking all constant terms to one side and all terms containing \[x\] to the other sides and then simplify the equation till we get the required result.

Complete step-by-step solution:
A linear inequality is an inequality in one variable that can be written in one of the following forms where \[a\] and \[b\] are real numbers and \[a \ne 0\],
\[ax + b < 0;ax + b > 0;ax + b \geqslant 0;ax + b \leqslant 0\].
Now given inequality is $ - 7x \geqslant 56$,
This is inequality in form \[ax + b \geqslant 0\], where \[a\] and \[b\] are constants,
Now taking the constant term to the other side we get,
\[ \Rightarrow ax \geqslant - b\],
Now taking all constant terms to one side we get,
\[ \Rightarrow x \geqslant - \dfrac{b}{a}\],
If a is positive the sign doesn’t change, and as equal to sign is there in the inequality one bracket will be closed bracket ,so solution for the above will be written as , \[x \in \left[ {\dfrac{{ - b}}{a},\infty } \right)\],
Now if a is negative then the sign changes, the equation becomes,
\[ \Rightarrow x \leqslant \dfrac{{ - b}}{a}\],
As equal to sign is there in the inequality one bracket will be closed bracket ,so solution for the above will be written as \[x \in \left( { - \infty ,\dfrac{{ - b}}{a}} \right]\].
Now the given inequality is $ - 7x \geqslant 56$,
Now divide both sides with 7, we get,
\[ \Rightarrow - \dfrac{{7x}}{7} \geqslant \dfrac{{56}}{7}\],
Now simplify the equation we get,
\[ \Rightarrow - x \geqslant 8\],
Now as \[x\] is having a negative sign we will multiply both sides with negative sign then the greater than symbol will change to less than,
\[ \Rightarrow - \left( { - x} \right) \leqslant - 8\],
Again simplifying we get,
\[ \Rightarrow x \leqslant - 8\],
This can also be written as \[\left( { - \infty , - 8} \right]\].

The solution of the given inequality $ - 7x \geqslant 56$ is \[\left( { - \infty , - 8} \right]\].

Note: There are three ways to represent solutions to inequalities: an interval, a graph, and an inequality. Because there is usually more than one solution to an inequality, when you check your answer you should check the end point and one other value to check the direction of the inequality. When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities.