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Formula
Solve the inequality
Graph
$\dfrac { 2 \left ( x - 1 \right ) } { 3 } - \dfrac { x - 1 } { 6 } \leq \dfrac { 2 } { 3 } x - 1$
$\dfrac { 2 \left ( x - 1 \right ) } { 3 } - \dfrac { x - 1 } { 6 } \leq \dfrac { 2 } { 3 } x - 1$
Solution of inequality
$x \geq 3$
$\dfrac{ 2 \left( x-1 \right) }{ 3 } - \dfrac{ x-1 }{ 6 } \leq \dfrac{ 2 }{ 3 } x-1$
$x \geq 3$
 Solve a solution to $x$
$\dfrac { \color{#FF6800}{ 2 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \right ) } { 3 } - \dfrac { x - 1 } { 6 } \leq \dfrac { 2 } { 3 } x - 1$
 Multiply each term in parentheses by $2$
$\dfrac { \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 2 } } { 3 } - \dfrac { x - 1 } { 6 } \leq \dfrac { 2 } { 3 } x - 1$
$\color{#FF6800}{ \dfrac { 2 x - 2 } { 3 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { x - 1 } { 6 } } \leq \dfrac { 2 } { 3 } x - 1$
 Write all numerators above the least common denominator 
$\color{#FF6800}{ \dfrac { 4 x - 4 - x + 1 } { 6 } } \leq \dfrac { 2 } { 3 } x - 1$
$\dfrac { \color{#FF6800}{ 4 } \color{#FF6800}{ x } - 4 \color{#FF6800}{ - } \color{#FF6800}{ x } + 1 } { 6 } \leq \dfrac { 2 } { 3 } x - 1$
 Calculate between similar terms 
$\dfrac { \color{#FF6800}{ 3 } \color{#FF6800}{ x } - 4 + 1 } { 6 } \leq \dfrac { 2 } { 3 } x - 1$
$\dfrac { 3 x \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ + } \color{#FF6800}{ 1 } } { 6 } \leq \dfrac { 2 } { 3 } x - 1$
 Add $- 4$ and $1$
$\dfrac { 3 x \color{#FF6800}{ - } \color{#FF6800}{ 3 } } { 6 } \leq \dfrac { 2 } { 3 } x - 1$
$\dfrac { 3 x - 3 } { 6 } \leq \color{#FF6800}{ \dfrac { 2 } { 3 } } \color{#FF6800}{ x } - 1$
 Calculate the multiplication expression 
$\dfrac { 3 x - 3 } { 6 } \leq \color{#FF6800}{ \dfrac { 2 x } { 3 } } - 1$
$\color{#FF6800}{ \dfrac { 3 x - 3 } { 6 } } \leq \color{#FF6800}{ \dfrac { 2 x } { 3 } } \color{#FF6800}{ - } \color{#FF6800}{ 1 }$
 Multiply both sides by the least common multiple for the denominators to eliminate the fraction 
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 3 } \leq \color{#FF6800}{ 4 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 6 }$
$3 x - 3 \leq \color{#FF6800}{ 4 } \color{#FF6800}{ x } - 6$
 Move the variable to the left-hand side and change the symbol 
$3 x - 3 \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \leq - 6$
$3 x \color{#FF6800}{ - } \color{#FF6800}{ 3 } - 4 x \leq - 6$
 Move the constant to the right side and change the sign 
$3 x - 4 x \leq - 6 \color{#FF6800}{ + } \color{#FF6800}{ 3 }$
$\color{#FF6800}{ 3 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \leq - 6 + 3$
 Organize the expression 
$\color{#FF6800}{ - } \color{#FF6800}{ x } \leq - 6 + 3$
$- x \leq \color{#FF6800}{ - } \color{#FF6800}{ 6 } \color{#FF6800}{ + } \color{#FF6800}{ 3 }$
 Add $- 6$ and $3$
$- x \leq \color{#FF6800}{ - } \color{#FF6800}{ 3 }$
$\color{#FF6800}{ - } \color{#FF6800}{ x } \leq \color{#FF6800}{ - } \color{#FF6800}{ 3 }$
 Change the symbol of the inequality of both sides, and reverse the symbol of the inequality to the opposite direction 
$\color{#FF6800}{ x } \geq \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 3 } \right )$
$x \geq \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 3 \right )$
 Simplify Minus 
$x \geq 3$
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