- #1
anemone
Gold Member
MHB
POTW Director
- 3,883
- 115
Show that the equation $|4x-5|-|3x+1|+|5-x|+|1+x|=0.99$ has no solutions.
Is the Absolute Value Inequality $|4x-5|-|3x+1|+|5-x|+|1+x|=0.99 Solvable?
- MHB
- Thread starter anemone
- Start date
In summary, "Absolute Value Inequality" is a mathematical inequality that represents the distance between a number and zero on a number line. To solve absolute value inequalities, the absolute value expression must be isolated and two cases must be considered. The main difference between absolute value equations and inequalities is the presence of an inequality sign. Absolute value inequalities are important in representing real-world scenarios and common mistakes include not considering both cases and incorrectly removing the absolute value bars.
- #2
MarkFL
Gold Member
MHB
- 13,288
- 12
My solution:
Let:
\(\displaystyle f(x)=|4x-5|-|3x+1|+|5-x|+|1+x|\)
We find that we may also write:
\(\displaystyle f(x)=\begin{cases}-3x+10, & x<-1 \\[3pt] -x+12, & -1\le x<-\dfrac{1}{3} \\[3pt] -7x+10, & -\dfrac{1}{3}\le x<\dfrac{5}{4} \\[3pt] x, & \dfrac{5}{4}\le x<5 \\[3pt] 3x-10, & 5\le x \\ \end{cases}\)
The graph of $f$ will have its minimum where the slope goes from negative to positive, thus we may conclude:
\(\displaystyle f_{\min}=f\left(\frac{5}{4}\right)=\frac{5}{4}\)
Hence:
\(\displaystyle f(x)=0.99\)
will have no real solution.
\(\displaystyle f(x)=|4x-5|-|3x+1|+|5-x|+|1+x|\)
We find that we may also write:
\(\displaystyle f(x)=\begin{cases}-3x+10, & x<-1 \\[3pt] -x+12, & -1\le x<-\dfrac{1}{3} \\[3pt] -7x+10, & -\dfrac{1}{3}\le x<\dfrac{5}{4} \\[3pt] x, & \dfrac{5}{4}\le x<5 \\[3pt] 3x-10, & 5\le x \\ \end{cases}\)
The graph of $f$ will have its minimum where the slope goes from negative to positive, thus we may conclude:
\(\displaystyle f_{\min}=f\left(\frac{5}{4}\right)=\frac{5}{4}\)
Hence:
\(\displaystyle f(x)=0.99\)
will have no real solution.
- #3
anemone
Gold Member
MHB
POTW Director
- 3,883
- 115
Good job, MarkFL! And thanks for participating! 
What is "Absolute Value Inequality"?
"Absolute Value Inequality" is a type of mathematical inequality that represents the distance between a number and zero on a number line. It is denoted by vertical bars surrounding the number and is always equal to or greater than zero.
How do you solve absolute value inequalities?
To solve absolute value inequalities, you must first isolate the absolute value expression on one side of the inequality. Then, you must consider two cases: when the expression inside the absolute value is positive and when it is negative. For each case, you can solve the inequality by removing the absolute value bars and solving for the variable.
What is the difference between absolute value equations and inequalities?
The main difference between absolute value equations and inequalities is that equations have an equal sign, while inequalities have an inequality sign (such as <, >, ≤, ≥). Additionally, while equations have one unique solution, inequalities can have multiple solutions or a range of solutions.
Why is it important to use absolute value inequalities?
Absolute value inequalities are important because they represent real-world scenarios where there is a distance or difference between two values. For example, they can be used to represent temperature changes, financial gains or losses, or any situation where the magnitude of a quantity is important.
What are some common mistakes when solving absolute value inequalities?
Some common mistakes when solving absolute value inequalities include forgetting to consider both cases, incorrectly removing the absolute value bars, and forgetting to flip the inequality sign when multiplying or dividing by a negative number. It is important to carefully follow the rules of solving inequalities to avoid these mistakes.
Similar threads
-
General Math
-
General Math
-
General Math
-
General Math
-
General Math
-
General Math
-
Precalculus Mathematics Homework Help
-
General Math
-
General Math
-
General Math