Question

How do you solve $2x + 7 \ge - 3$?

Answer 1

Hint: We will have to solve for $x$ simply by solving the above equation, not just as an equation but with keeping inequality in mind, i.e., the addition or subtraction of variables and constants on both sides rather than simply transposing them. First, move the constant part on the right side of the inequality. After that divide both sides by the coefficient of $x$ to solve for $x$.

Complete step-by-step answer:
A linear inequality is a mathematical statement that relates a linear expression as either less than or greater than another.
A solution to a linear inequality is a real number that will produce a true statement when substituted for the variable. Linear inequalities have either infinitely many solutions or no solution. If there are infinitely many solutions, graph the solution set on a number line and/or express the solution using interval notation.
Here, we have the given inequality of the form
$ \Rightarrow 2x + 7 \ge - 3$
In the above equation, we could have simply transposed the RHS variables to LHS but that is inappropriate for an inequality-based equation.
So, we have to add or subtract the variables on both sides to cancel out on one side,
By subtracting 7 on both sides, we get,
$ \Rightarrow 2x + 7 - 7 \ge - 3 - 7$
Simplify the terms,
$ \Rightarrow 2x \ge - 10$
Now, dividing by 2 into both sides, we get
$ \Rightarrow \dfrac{{2x}}{2} \ge - \dfrac{{10}}{2}$
Cancel out the common factor,
$ \Rightarrow x \ge - 5$

Hence, we can say that the value of real $x$ with given inequality is $x \in \left[ { - 5,\infty } \right)$.

Note:
A simple mistake that is very common in this kind of problem is, students generally transpose the variables across the inequality like a normal equation which is not preferred, especially in the case of multiplications and divisions.