Solving 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 \[-5x + 5 \ge -x {\,+\,}1\] via equations.
\(x \le 1\)
We follow the following roadmap:
- Get started with the corresponding equation \[-5x + 5 = -x {\,+\,}1\]
- Solve this equation:
- Get the terms with \(x\) on the left-hand side of the equation (by adding \(x\) on both sides):
\(-5x + 5 +x = -x {\,+\,}1 +x\), which simplifies to \(-4x +5 = 1\). - Then move the terms without \(x\) to the right (by adding \(-5\) both sides):
\(-4x +5 - 5 = 1 - 5\), which simplifies to \(-4x = -4\).- Next, divide the left- and right-hand side by the coefficient of \(x\) (which is here \(-4\)); this gives \(x = \;\frac{-4}{-4}\).
- So, the solution of the equation is \(x = {1}\).
- Get the terms with \(x\) on the left-hand side of the equation (by adding \(x\) 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 \(-5x + 5 \ge -x {\,+\,}1\) when you substitute a value of \(x\) less than or equal to \(1\). For example, when you fill in \(x=-10\), then you get \(55 \ge 11\) 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 \(-5x + 5 \ge -x {\,+\,}1\) when you substitute a value of \(x\) greater than or equal to \(1\). For example, when you fill in \(x=10\), then you get \(-45 \ge -9\) 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 \(-5x + 5 \ge -x {\,+\,}1\) 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.
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