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Question

\left|4x+2\right|+\left|2x-5\right|=11
Solve the equation
x_{1}=-\frac{4}{3},x_{2}=2
Alternative Form
x_{1}=-1.\dot{3},x_{2}=2
Evaluate
\left|4x+2\right|+\left|2x-5\right|=11
Move the expression to the left side
\left|4x+2\right|+\left|2x-5\right|-11=0
\text{Separate the equation into }4\text{ possible cases}
\begin{align}&4x+2+2x-5-11=0,4x+2\geq 0,2x-5\geq 0\\&4x+2-\left(2x-5\right)-11=0,4x+2\geq 0,2x-5<0\\&-\left(4x+2\right)+2x-5-11=0,4x+2<0,2x-5\geq 0\\&-\left(4x+2\right)-\left(2x-5\right)-11=0,4x+2<0,2x-5<0\end{align}
Solve the equation
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Evaluate
4x+2+2x-5-11=0
Calculate the sum or difference
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Evaluate
4x+2+2x-5-11
Add the terms
6x+2-5-11
Subtract the numbers
6x-14
6x-14=0
Move the constant to the right-hand side and change its sign
6x=0+14
Removing 0 doesn't change the value,so remove it from the expression
6x=14
Divide both sides
\frac{6x}{6}=\frac{14}{6}
Divide the numbers
x=\frac{14}{6}
\text{Cancel out the common factor }2
x=\frac{7}{3}
\begin{align}&x=\frac{7}{3},4x+2\geq 0,2x-5\geq 0\\&4x+2-\left(2x-5\right)-11=0,4x+2\geq 0,2x-5<0\\&-\left(4x+2\right)+2x-5-11=0,4x+2<0,2x-5\geq 0\\&-\left(4x+2\right)-\left(2x-5\right)-11=0,4x+2<0,2x-5<0\end{align}
Solve the inequality
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Evaluate
4x+2\geq 0
Move the constant to the right side
4x\geq 0-2
Removing 0 doesn't change the value,so remove it from the expression
4x\geq -2
Divide both sides
\frac{4x}{4}\geq \frac{-2}{4}
Divide the numbers
x\geq \frac{-2}{4}
Divide the numbers
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Evaluate
\frac{-2}{4}
\text{Cancel out the common factor }2
\frac{-1}{2}
\text{Use }\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\text{ to rewrite the fraction}
-\frac{1}{2}
x\geq -\frac{1}{2}
\begin{align}&x=\frac{7}{3},x\geq -\frac{1}{2},2x-5\geq 0\\&4x+2-\left(2x-5\right)-11=0,4x+2\geq 0,2x-5<0\\&-\left(4x+2\right)+2x-5-11=0,4x+2<0,2x-5\geq 0\\&-\left(4x+2\right)-\left(2x-5\right)-11=0,4x+2<0,2x-5<0\end{align}
Solve the inequality
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Evaluate
2x-5\geq 0
Move the constant to the right side
2x\geq 0+5
Removing 0 doesn't change the value,so remove it from the expression
2x\geq 5
Divide both sides
\frac{2x}{2}\geq \frac{5}{2}
Divide the numbers
x\geq \frac{5}{2}
\begin{align}&x=\frac{7}{3},x\geq -\frac{1}{2},x\geq \frac{5}{2}\\&4x+2-\left(2x-5\right)-11=0,4x+2\geq 0,2x-5<0\\&-\left(4x+2\right)+2x-5-11=0,4x+2<0,2x-5\geq 0\\&-\left(4x+2\right)-\left(2x-5\right)-11=0,4x+2<0,2x-5<0\end{align}
Solve the equation
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Evaluate
4x+2-\left(2x-5\right)-11=0
Calculate
4x+2-2x+5-11=0
Calculate the sum or difference
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Evaluate
4x+2-2x+5-11
Subtract the terms
2x+2+5-11
Calculate the sum or difference
2x-4
2x-4=0
Move the constant to the right-hand side and change its sign
2x=0+4
Removing 0 doesn't change the value,so remove it from the expression
2x=4
Divide both sides
\frac{2x}{2}=\frac{4}{2}
Divide the numbers
x=\frac{4}{2}
Divide the numbers
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Evaluate
\frac{4}{2}
Reduce the numbers
\frac{2}{1}
Calculate
2
x=2
\begin{align}&x=\frac{7}{3},x\geq -\frac{1}{2},x\geq \frac{5}{2}\\&x=2,4x+2\geq 0,2x-5<0\\&-\left(4x+2\right)+2x-5-11=0,4x+2<0,2x-5\geq 0\\&-\left(4x+2\right)-\left(2x-5\right)-11=0,4x+2<0,2x-5<0\end{align}
Solve the inequality
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Evaluate
4x+2\geq 0
Move the constant to the right side
4x\geq 0-2
Removing 0 doesn't change the value,so remove it from the expression
4x\geq -2
Divide both sides
\frac{4x}{4}\geq \frac{-2}{4}
Divide the numbers
x\geq \frac{-2}{4}
Divide the numbers
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Evaluate
\frac{-2}{4}
\text{Cancel out the common factor }2
\frac{-1}{2}
\text{Use }\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\text{ to rewrite the fraction}
-\frac{1}{2}
x\geq -\frac{1}{2}
\begin{align}&x=\frac{7}{3},x\geq -\frac{1}{2},x\geq \frac{5}{2}\\&x=2,x\geq -\frac{1}{2},2x-5<0\\&-\left(4x+2\right)+2x-5-11=0,4x+2<0,2x-5\geq 0\\&-\left(4x+2\right)-\left(2x-5\right)-11=0,4x+2<0,2x-5<0\end{align}
Solve the inequality
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Evaluate
2x-5<0
Move the constant to the right side
2x<0+5
Removing 0 doesn't change the value,so remove it from the expression
2x<5
Divide both sides
\frac{2x}{2}<\frac{5}{2}
Divide the numbers
x<\frac{5}{2}
\begin{align}&x=\frac{7}{3},x\geq -\frac{1}{2},x\geq \frac{5}{2}\\&x=2,x\geq -\frac{1}{2},x<\frac{5}{2}\\&-\left(4x+2\right)+2x-5-11=0,4x+2<0,2x-5\geq 0\\&-\left(4x+2\right)-\left(2x-5\right)-11=0,4x+2<0,2x-5<0\end{align}
Solve the equation
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Evaluate
-\left(4x+2\right)+2x-5-11=0
Calculate
-4x-2+2x-5-11=0
Calculate the sum or difference
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Evaluate
-4x-2+2x-5-11
Add the terms
-2x-2-5-11
Subtract the numbers
-2x-18
-2x-18=0
Move the constant to the right-hand side and change its sign
-2x=0+18
Removing 0 doesn't change the value,so remove it from the expression
-2x=18
Change the signs on both sides of the equation
2x=-18
Divide both sides
\frac{2x}{2}=\frac{-18}{2}
Divide the numbers
x=\frac{-18}{2}
Divide the numbers
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Evaluate
\frac{-18}{2}
Reduce the numbers
\frac{-9}{1}
Calculate
-9
x=-9
\begin{align}&x=\frac{7}{3},x\geq -\frac{1}{2},x\geq \frac{5}{2}\\&x=2,x\geq -\frac{1}{2},x<\frac{5}{2}\\&x=-9,4x+2<0,2x-5\geq 0\\&-\left(4x+2\right)-\left(2x-5\right)-11=0,4x+2<0,2x-5<0\end{align}
Solve the inequality
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Evaluate
4x+2<0
Move the constant to the right side
4x<0-2
Removing 0 doesn't change the value,so remove it from the expression
4x<-2
Divide both sides
\frac{4x}{4}<\frac{-2}{4}
Divide the numbers
x<\frac{-2}{4}
Divide the numbers
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Evaluate
\frac{-2}{4}
\text{Cancel out the common factor }2
\frac{-1}{2}
\text{Use }\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\text{ to rewrite the fraction}
-\frac{1}{2}
x<-\frac{1}{2}
\begin{align}&x=\frac{7}{3},x\geq -\frac{1}{2},x\geq \frac{5}{2}\\&x=2,x\geq -\frac{1}{2},x<\frac{5}{2}\\&x=-9,x<-\frac{1}{2},2x-5\geq 0\\&-\left(4x+2\right)-\left(2x-5\right)-11=0,4x+2<0,2x-5<0\end{align}
Solve the inequality
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Evaluate
2x-5\geq 0
Move the constant to the right side
2x\geq 0+5
Removing 0 doesn't change the value,so remove it from the expression
2x\geq 5
Divide both sides
\frac{2x}{2}\geq \frac{5}{2}
Divide the numbers
x\geq \frac{5}{2}
\begin{align}&x=\frac{7}{3},x\geq -\frac{1}{2},x\geq \frac{5}{2}\\&x=2,x\geq -\frac{1}{2},x<\frac{5}{2}\\&x=-9,x<-\frac{1}{2},x\geq \frac{5}{2}\\&-\left(4x+2\right)-\left(2x-5\right)-11=0,4x+2<0,2x-5<0\end{align}
Solve the equation
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Evaluate
-\left(4x+2\right)-\left(2x-5\right)-11=0
Calculate
-4x-2-2x+5-11=0
Calculate the sum or difference
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Evaluate
-4x-2-2x+5-11
Subtract the terms
-6x-2+5-11
Calculate the sum or difference
-6x-8
-6x-8=0
Move the constant to the right-hand side and change its sign
-6x=0+8
Removing 0 doesn't change the value,so remove it from the expression
-6x=8
Change the signs on both sides of the equation
6x=-8
Divide both sides
\frac{6x}{6}=\frac{-8}{6}
Divide the numbers
x=\frac{-8}{6}
Divide the numbers
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Evaluate
\frac{-8}{6}
\text{Cancel out the common factor }2
\frac{-4}{3}
\text{Use }\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\text{ to rewrite the fraction}
-\frac{4}{3}
x=-\frac{4}{3}
\begin{align}&x=\frac{7}{3},x\geq -\frac{1}{2},x\geq \frac{5}{2}\\&x=2,x\geq -\frac{1}{2},x<\frac{5}{2}\\&x=-9,x<-\frac{1}{2},x\geq \frac{5}{2}\\&x=-\frac{4}{3},4x+2<0,2x-5<0\end{align}
Solve the inequality
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Evaluate
4x+2<0
Move the constant to the right side
4x<0-2
Removing 0 doesn't change the value,so remove it from the expression
4x<-2
Divide both sides
\frac{4x}{4}<\frac{-2}{4}
Divide the numbers
x<\frac{-2}{4}
Divide the numbers
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Evaluate
\frac{-2}{4}
\text{Cancel out the common factor }2
\frac{-1}{2}
\text{Use }\frac{-a}{b}=\frac{a}{-b}=-\frac{a}{b}\text{ to rewrite the fraction}
-\frac{1}{2}
x<-\frac{1}{2}
\begin{align}&x=\frac{7}{3},x\geq -\frac{1}{2},x\geq \frac{5}{2}\\&x=2,x\geq -\frac{1}{2},x<\frac{5}{2}\\&x=-9,x<-\frac{1}{2},x\geq \frac{5}{2}\\&x=-\frac{4}{3},x<-\frac{1}{2},2x-5<0\end{align}
Solve the inequality
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Evaluate
2x-5<0
Move the constant to the right side
2x<0+5
Removing 0 doesn't change the value,so remove it from the expression
2x<5
Divide both sides
\frac{2x}{2}<\frac{5}{2}
Divide the numbers
x<\frac{5}{2}
\begin{align}&x=\frac{7}{3},x\geq -\frac{1}{2},x\geq \frac{5}{2}\\&x=2,x\geq -\frac{1}{2},x<\frac{5}{2}\\&x=-9,x<-\frac{1}{2},x\geq \frac{5}{2}\\&x=-\frac{4}{3},x<-\frac{1}{2},x<\frac{5}{2}\end{align}
Find the intersection
\begin{align}&x \in \varnothing \\&x=2,x\geq -\frac{1}{2},x<\frac{5}{2}\\&x=-9,x<-\frac{1}{2},x\geq \frac{5}{2}\\&x=-\frac{4}{3},x<-\frac{1}{2},x<\frac{5}{2}\end{align}
Find the intersection
\begin{align}&x \in \varnothing \\&x=2\\&x=-9,x<-\frac{1}{2},x\geq \frac{5}{2}\\&x=-\frac{4}{3},x<-\frac{1}{2},x<\frac{5}{2}\end{align}
Find the intersection
\begin{align}&x \in \varnothing \\&x=2\\&x \in \varnothing \\&x=-\frac{4}{3},x<-\frac{1}{2},x<\frac{5}{2}\end{align}
Find the intersection
\begin{align}&x \in \varnothing \\&x=2\\&x \in \varnothing \\&x=-\frac{4}{3}\end{align}
Find the union
\begin{align}&x=2\\&x=-\frac{4}{3}\end{align}
Solution
x_{1}=-\frac{4}{3},x_{2}=2
Alternative Form
x_{1}=-1.\dot{3},x_{2}=2
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