Equations and inequalities with absolute values
Definition of the absolute value function
Absolute value function
The following piecewise function is the absolute value function:
|x|=\begin{cases} x & x\geq 0 \cr \cr -x & x\leq 0 \end{cases}
Consider the absolute value function:
f\left(x\right)=|x|
Then we have the following:
- |3|=3
- |-3|=-\left(-3\right)=3
In practice, the absolute value function ignores any negatives.
Equations with absolute values
Let a be a real number and consider the following equation:
|x|=a
Then:
- If a\geq 0, the two solutions are x= a and x=-a.
- If a \lt 0, the equation has no solutions.
Consider the following equation:
|x|=3
Then the solutions to the above equation are x=3 and x=-3.
Consider the following equation:
|x|=-2
Then the equation has no solutions.
The above property for equations involving the absolute value function allows us to solve more complicated equations.
Consider the following equation:
|2x+7|=11
Then the above equation is true whenever:
\begin{cases} 2x+7=11\cr \cr \text{or} \cr \cr2x+7=-11 \end{cases}
The first equation is solved as follows:
2x+7=11\\2x=4\\x=2
The second equation is solved as follows:
2x+7=-11\\2x=-18\\x=-9
Therefore the solutions to the original equation are x=2 and x=-9.
Suppose f\left(x\right) and g\left(x\right) are functions, then |f\left(x\right)|=|g\left(x\right)| if and only if:
\begin{cases} f\left(x\right)=g\left(x\right) \cr \cr \mbox{or} \cr \cr f\left(x\right)=-g\left(x\right) \end{cases}
Consider the following equation:
|2x+7|=|3x+9|
Then the above equation is true whenever
\begin{cases} 2x+7=3x+9 \cr \cr \text{or} \cr \cr 2x+7=-\left(3x+9\right) \end{cases}
The first equation is solved as follows:
2x+7=3x+9\\7=x+9\\x=-2
The second equation is solved as follows:
2x+7=-\left(3x+9\right)\\2x+7=-3x-9\\5x=-16\\x=\dfrac{-16}{5}
Therefore, the solutions to the original equation are x=-2 and x=\dfrac{-16}{5}.
Inequalities with absolute values
Let a be a real number and consider the following inequality:
|x|\leq a
Then:
- If a\geq 0, the solution set is \left[-a,a\right].
- If a \lt 0, the equation has no solutions.
Consider the following inequality:
|x|\leq 3
Then the solution set to the inequality is:
\left[-3{,}3\right]
This can also be seen graphically by graphing the absolute value function and shading everything below the line y=3. The portion of the graph of the absolute value function lying in the shaded region corresponds to solutions to the inequality.
Let a be a real number and consider the following inequality:
|x|\geq a
Then:
- If a\geq 0, then the solution set is \left(-\infty, -a\left]\cup \right[a,\infty\right).
- If a \lt 0, then the solution set is all real numbers.
Consider the following inequality:
|x|\geq 2
Then the solution set to the inequality is
\left(-\infty, -2\left]\cup \right[2, \infty\right)
This can also be seen graphically by graphing the absolute value function and shading everything above the line y=2. The portion of the graph of the absolute value function lying in the shaded region corresponds to solutions to the inequality.
The above property of inequalities involving the absolute value function allows us to solve more complicated inequalities.
Consider the following inequality:
|2x+7|\leq 13
Then the inequality is equivalent to the following:
-13\leq 2x+7\leq 13\\-20\leq 2x\leq 6\\-10\leq x\leq 3
Therefore the solution set to the inequality is \left[-10, 3\right].
Equations and inequalities with greatest integer function
Definition of the greatest integer function
Greatest integer function
For each real number a let \lfloor a \rfloor be the greatest integer less than or equal to a. The greatest integer function is defined to be
f\left(x\right)=\lfloor x \rfloor
The following graph is the graph of the greatest integer function.
Consider the greatest integer function:
f\left(x\right)=\lfloor x \rfloor
Then we have the following:
- f\left(3.1\right)=\lfloor 3.1 \rfloor =3
- f\left(-3.1\right)=\lfloor -3.1 \rfloor =-4
- f\left(3\right)=\lfloor 3 \rfloor =3
Equations with greatest integer function
Let a be a real number and consider the following equation:
\lfloor x \rfloor =a
Then:
- If a is an integer, the solution set is \left[a,a+1\right).
- If a is not an integer, the equation has no solution.
Consider the following equation:
\lfloor x \rfloor =2
Then the solution set to the equation is
\left[2{,}3\right)
This is also seen graphically by graphing the greatest integer function and the line y=2. The portion of the graph of the greatest integer function lying on the line y=2 correspond to solutions to the equation.
Consider the following equation:
\lfloor x \rfloor =2.1
Then the equation has no solutions.
This is also seen graphically by graphing the greatest integer function and the line y=2.1. No portion of the graph of the greatest integer function intersects the line y=2.1 and therefore there are no solutions to the above equation.
The above property of the greatest integer function allows us to solve more complicated equations.
Consider the following equation:
\lfloor 2x-5\rfloor =12
Then the equation is equivalent to the following:
12\leq 2x-5\lt 13\\17\leq 2x\lt 18\\\dfrac{17}{2}\leq x\lt 9
Therefore the solution set to the equation is \left[\dfrac{17}{2},9\right).
Inequalities with the greatest integer function
Let a be a real number and consider the following inequality:
\lfloor x \rfloor \leq a
Then the solution set to the inequality is:
\left(-\infty, \lfloor a\rfloor+1\right)
Consider the following inequality:
\lfloor x \rfloor \leq 2.25
Then the solution set to the inequality is:
\left(-\infty, 3\right)
This can also be seen graphically by graphing the greatest integer function and shading everything below the line y=2.25. The portions of the graph of the greatest integer function correspond to the solutions to the inequality.
Let a be a real number and consider the following inequalitiy:
\lfloor x \rfloor \lt a
Then:
- If a is an integer, the solution set is \left(-\infty, a\right).
- If a is not an integer, the solution set is \left(-\infty, \lfloor a\rfloor+1\right) .
Consider the following inequality:
\lfloor 2x+1 \rfloor \lt 3
Then the inequality is equivalent to the following:
2x+1 \lt 3\\2x \lt 2\\x \lt 1
Therefore the solution set to the inequality is \left(-\infty, 1\right).
Consider the following inequality:
\lfloor 2x+1\rfloor \lt 3.1
Then the inequality is equivalent to the following:
2x+1\lt \lfloor 3.1 \rfloor +1\\2x+1 \lt 4\\2x \lt 3\\x \lt \dfrac{3}{2}
Therefore the inequality has solution set \left(-\infty, \dfrac{3}{2}\right).
Let a be a real number and consider the following inequality:
\lfloor x\rfloor \gt a
Then the solution set to the inequality is
\left[\lfloor a \rfloor +1, \infty\right)
Consider the following inequality:
\lfloor x \rfloor \gt 1
Then the solution set to the inequality is
\left[2, \infty\right)
This can be seen graphically by graphing the greatest integer function, shading everything above the line y=1. Then the portion of the graph lying strictly in the shaded region corresponds to solutions to the inequality.
Let a be a real number and consider the following inequality:
\lfloor x \rfloor \geq a
Then the solution set to the inequality is:
- \left[a, \infty\right) if a is an integer.
- \left[\lfloor a \rfloor +1, \infty \right) if a is not an integer.
Consider the following inequality:
\lfloor 2x+1\rfloor \geq 1
Then the inequality is equivalent to the following:
2x+1\geq 1\\2x\geq 0\\x\geq 0
Therefore the solution set to the original inequality is \left[0,\infty\right).
Consider the following inequality:
\lfloor 2x+1\rfloor \geq 1.1
Then the inequality is equivalent to the following:
2x+1\geq \lfloor 1.1 \rfloor +1\\2x+1\geq 2\\2x\geq 1\\x\geq \dfrac{1}{2}
Therefore the solution set to the inequality is \left[\dfrac{1}{2}, \infty\right).