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Formula
Solve the inequality
Graph
$\dfrac { 1 } { 3 } x + 0.5 \left ( x - \dfrac { 1 } { 3 } \right ) \geq \dfrac { 3 } { 2 } x$
$\dfrac { 1 } { 3 } x + 0.5 \left ( x - \dfrac { 1 } { 3 } \right ) \geq \dfrac { 3 } { 2 } x$
Solution of inequality
$x \leq - \dfrac { 1 } { 4 }$
$\dfrac{ 1 }{ 3 } x+0.5 \left( x- \dfrac{ 1 }{ 3 } \right) \geq \dfrac{ 3 }{ 2 } x$
$x \leq - \dfrac { 1 } { 4 }$
 Solve a solution to $x$
$\dfrac { 1 } { 3 } x + \color{#FF6800}{ 0.5 } \left ( \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 3 } } \right ) \geq \dfrac { 3 } { 2 } x$
 Multiply each term in parentheses by $0.5$
$\dfrac { 1 } { 3 } x + \color{#FF6800}{ 0.5 } \color{#FF6800}{ x } + \color{#FF6800}{ 0.5 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 3 } } \right ) \geq \dfrac { 3 } { 2 } x$
$\color{#FF6800}{ \dfrac { 1 } { 3 } } \color{#FF6800}{ x } + 0.5 x + 0.5 \times \left ( - \dfrac { 1 } { 3 } \right ) \geq \dfrac { 3 } { 2 } x$
 Calculate the multiplication expression 
$\color{#FF6800}{ \dfrac { x } { 3 } } + 0.5 x + 0.5 \times \left ( - \dfrac { 1 } { 3 } \right ) \geq \dfrac { 3 } { 2 } x$
$\dfrac { x } { 3 } + \color{#FF6800}{ 0.5 } \color{#FF6800}{ x } + 0.5 \times \left ( - \dfrac { 1 } { 3 } \right ) \geq \dfrac { 3 } { 2 } x$
 Calculate the multiplication expression 
$\dfrac { x } { 3 } + \color{#FF6800}{ \dfrac { x } { 2 } } + 0.5 \times \left ( - \dfrac { 1 } { 3 } \right ) \geq \dfrac { 3 } { 2 } x$
$\dfrac { x } { 3 } + \dfrac { x } { 2 } + \color{#FF6800}{ 0.5 } \color{#FF6800}{ \times } \left ( \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 3 } } \right ) \geq \dfrac { 3 } { 2 } x$
 Calculate multiplication and division of rational numbers 
$\dfrac { x } { 3 } + \dfrac { x } { 2 } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 6 } } \geq \dfrac { 3 } { 2 } x$
$\color{#FF6800}{ \dfrac { x } { 3 } } \color{#FF6800}{ + } \color{#FF6800}{ \dfrac { x } { 2 } } - \dfrac { 1 } { 6 } \geq \dfrac { 3 } { 2 } x$
 Write all numerators above the least common denominator 
$\color{#FF6800}{ \dfrac { 2 x + 3 x } { 6 } } - \dfrac { 1 } { 6 } \geq \dfrac { 3 } { 2 } x$
$\color{#FF6800}{ \dfrac { 2 x + 3 x } { 6 } } \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 6 } } \geq \dfrac { 3 } { 2 } x$
 Since the denominator is the same as $6$ , combine the fractions into one 
$\color{#FF6800}{ \dfrac { 2 x + 3 x - 1 } { 6 } } \geq \dfrac { 3 } { 2 } x$
$\dfrac { \color{#FF6800}{ 2 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 3 } \color{#FF6800}{ x } - 1 } { 6 } \geq \dfrac { 3 } { 2 } x$
 Calculate between similar terms 
$\dfrac { \color{#FF6800}{ 5 } \color{#FF6800}{ x } - 1 } { 6 } \geq \dfrac { 3 } { 2 } x$
$\dfrac { 5 x - 1 } { 6 } \geq \color{#FF6800}{ \dfrac { 3 } { 2 } } \color{#FF6800}{ x }$
 Calculate the multiplication expression 
$\dfrac { 5 x - 1 } { 6 } \geq \color{#FF6800}{ \dfrac { 3 x } { 2 } }$
$\color{#FF6800}{ \dfrac { 5 x - 1 } { 6 } } \geq \color{#FF6800}{ \dfrac { 3 x } { 2 } }$
 Multiply both sides by the least common multiple for the denominators to eliminate the fraction 
$\color{#FF6800}{ 5 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 1 } \geq \color{#FF6800}{ 9 } \color{#FF6800}{ x }$
$5 x - 1 \geq \color{#FF6800}{ 9 } \color{#FF6800}{ x }$
 Move the variable to the left-hand side and change the symbol 
$5 x - 1 \color{#FF6800}{ - } \color{#FF6800}{ 9 } \color{#FF6800}{ x } \geq 0$
$5 x \color{#FF6800}{ - } \color{#FF6800}{ 1 } - 9 x \geq 0$
 Move the constant to the right side and change the sign 
$5 x - 9 x \geq \color{#FF6800}{ 1 }$
$\color{#FF6800}{ 5 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 9 } \color{#FF6800}{ x } \geq 1$
 Organize the expression 
$\color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \geq 1$
$\color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ x } \geq \color{#FF6800}{ 1 }$
 Change the symbol of the inequality of both sides, and reverse the symbol of the inequality to the opposite direction 
$4 x \leq - 1$
$\color{#FF6800}{ 4 } \color{#FF6800}{ x } \leq \color{#FF6800}{ - } \color{#FF6800}{ 1 }$
 Divide both sides by the same number 
$\color{#FF6800}{ x } \leq \color{#FF6800}{ - } \color{#FF6800}{ \dfrac { 1 } { 4 } }$
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