These statements also apply to
$\text{\hspace{0.17em}}\left|X\right|\le k\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}\left|X\right|\ge k.$
Determining a number within a prescribed distance
Describe all values
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ within a distance of 4 from the number 5.
We want the distance between
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and 5 to be less than or equal to 4. We can draw a number line, such as in
[link], to represent the condition to be satisfied.
The distance from
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ to 5 can be represented using an absolute value symbol,
$\text{\hspace{0.17em}}\left|x-5\right|.\text{\hspace{0.17em}}$ Write the values of
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ that satisfy the condition as an absolute value inequality.
$\left|x-5\right|\le 4$
We need to write two inequalities as there are always two solutions to an absolute value equation.
If the solution set is
$\text{\hspace{0.17em}}x\le 9\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}x\ge 1,$ then the solution set is an interval including all real numbers between and including 1 and 9.
So
$\text{\hspace{0.17em}}\left|x-5\right|\le 4\text{\hspace{0.17em}}$ is equivalent to
$\text{\hspace{0.17em}}\left[1,9\right]\text{\hspace{0.17em}}$ in interval notation.
Using a graphical approach to solve absolute value inequalities
Given the equation
$y=-\frac{1}{2}|4x-5|+3,$ determine the
x -values for which the
y -values are negative.
We are trying to determine where
$\text{\hspace{0.17em}}y<0,$ which is when
$\text{\hspace{0.17em}}-\frac{1}{2}|4x-5|+3<0.\text{\hspace{0.17em}}$ We begin by isolating the absolute value.
Now, we can examine the graph to observe where the
y- values are negative. We observe where the branches are below the
x- axis. Notice that it is not important exactly what the graph looks like, as long as we know that it crosses the horizontal axis at
$\text{\hspace{0.17em}}x=-\frac{1}{4}\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}x=\frac{11}{4},$ and that the graph opens downward. See
[link].
$k\le 1\text{\hspace{0.17em}}$ or
$\text{\hspace{0.17em}}k\ge 7;$ in interval notation, this would be
$\text{\hspace{0.17em}}(-\infty ,1]\cup [7,\infty ).$
Interval notation is a method to indicate the solution set to an inequality. Highly applicable in calculus, it is a system of parentheses and brackets that indicate what numbers are included in a set and whether the endpoints are included as well. See
[link] and
[link].
Solving inequalities is similar to solving equations. The same algebraic rules apply, except for one: multiplying or dividing by a negative number reverses the inequality. See
[link],[link] ,
[link] , and
[link].
Compound inequalities often have three parts and can be rewritten as two independent inequalities. Solutions are given by boundary values, which are indicated as a beginning boundary or an ending boundary in the solutions to the two inequalities. See
[link] and
[link] .
Absolute value inequalities will produce two solution sets due to the nature of absolute value. We solve by writing two equations: one equal to a positive value and one equal to a negative value. See
[link] and
[link].
Absolute value inequalities can also be solved by graphing. At least we can check the algebraic solutions by graphing, as we cannot depend on a visual for a precise solution. See
[link] .
Questions & Answers
show that the set of all natural number form semi group under the composition of addition
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the
fraction, the value of the fraction becomes 2/3. Find the original fraction.
2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
Q2
x+(x+2)+(x+4)=60
3x+6=60
3x+6-6=60-6
3x=54
3x/3=54/3
x=18
:. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
Solve for the first variable in one of the equations, then substitute the result into the other equation.
Point For:
(6111,4111,−411)(6111,4111,-411)
Equation Form:
x=6111,y=4111,z=−411x=6111,y=4111,z=-411