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 \[-x + 3 \lt 2x -6\] via equations.
\(x \gt 3\)
We follow the following roadmap:
- Get started with the corresponding equation \[-x + 3 = 2x -6\]
- Solve this equation:
- Get the terms with \(x\) on the left-hand side of the equation (by adding \(-2x\) on both sides):
\(-x + 3 - 2x = 2x -6 - 2x\), which simplifies to \(-3x +3 = -6\). - Then move the terms without \(x\) to the right (by adding \(-3\) both sides):
\(-3x +3 - 3 = -6 - 3\), which simplifies to \(-3x = -9\).- Next, divide the left- and right-hand side by the coefficient of \(x\) (which is here \(-3\)); this gives \(x = \;\frac{-9}{-3}\).
- So, the solution of the equation is \(x = {3}\).
- 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 \(3\).
- First calculate the left- and right-hand sides of the inequality \(-x + 3 \lt 2x -6\) when you substitute a value of \(x\) less than \(3\). For example, when you fill in \(x=-10\), then you get \(13 \lt -26\) and this is a false statement. Any other value of \(x\) less than \(3\) may be used too, and you still get a false statement.
- Then calculate the left- and right-hand sides of the inequality \(-x + 3 \lt 2x -6\) when you substitute a value of \(x\) greater than \(3\). For example, when you fill in \(x=10\), then you get \(-7 \lt 14\) and this is a true statement. Any other value of \(x\) greater than \(3\) may be used too, and you still get a true statement.
- From these two numeric examples follows that solutions \(x\) of \(-x + 3 \lt 2x -6\) must satisfy \(x \gt 3\).
The points where the inequality holds are shown in green in the number line below. An open circle around \(x=3\) 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.
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