Solving linear equations and inequalities: Linear inequalities in one unknown
Solving a linear inequality via equations
You can also solve a linear inequality by
- first replacing the inequality sign by an equal sign,
- then solving this equation, and
- finally, determining the sign of the inequality for point to the left and to the right of the solution of the equation.
Determine the exact solution of the inequality \[-3x -6 \ge -2x -5\] via equations.
\(x \le -1\)
We follow the following roadmap:
- Get started with the corresponding equation \[-3x -6 = -2x -5\]
- Solve this equation:
- Get the terms with \(x\) on the left-hand side of the equation (by adding \(2x\) on both sides):
\(-3x -6 +2x = -2x -5 +2x\), which simplifies to \(-x -6 = -5\). - Then move the terms without \(x\) to the right (by adding \(6\) both sides):
\(-x -6 +6 = -5 +6\), which simplifies to \(-x = 1\).- Next, divide the left- and right-hand side by the coefficient of \(x\) (which is here \(-1\)); this gives \(x = \;\frac{1}{-1}\).
- So, the solution of the equation is \(x = {-1}\).
- Get the terms with \(x\) on the left-hand side of the equation (by adding \(2x\) on both sides):
- Find out whether the solutions are on the number line to the left or to the right of \(-1\).
- First calculate the left- and right-hand sides of the inequality \(-3x -6 \ge -2x -5\) when you substitute a value of \(x\) less than or equal to \(-1\). For example, when you fill in \(x=-10\), then you get \(24 \ge 15\) and this is a true statement. Any other value of \(x\) less than or equal to \(-1\) may be used too, and you still get a true statement.
- Then calculate the left- and right-hand sides of the inequality \(-3x -6 \ge -2x -5\) when you substitute a value of \(x\) greater than or equal to \(-1\). For example, when you fill in \(x=10\), then you get \(-36 \ge -25\) and this is a false statement. Any other value of \(x\) greater than or equal to \(-1\) may be used too, and you still get a false statement.
- From these two numeric examples follows that solutions \(x\) of \(-3x -6 \ge -2x -5\) must satisfy \(x \le -1\).
The points where the inequality holds are shown in green in the number line below. An open circle around \(x=-1\) indicates that we are dealing with an inequality of the type \(\lt\) or \(\gt\), where in this case the point itself is not a solution. A closed circle indicates an inequality of the type \(\le\) or \(\ge\), and then the point marked on the number line is element of the solution set.
Unlock full access