Question

Solve the inequality \[\dfrac{{4x}}{3} - \dfrac{9}{4} < x + \dfrac{3}{4}\], \[\dfrac{{7x - 1}}{3} - \dfrac{{7x + 2}}{6} > x\]
A). \[\left( { - \infty ,4} \right) \cup \left( {9,\infty } \right)\]
B). \[\left( {4,\left. 9 \right)} \right.\]
C). \[\left( { - \infty ,\infty } \right)\]
D). \[\left( { - \infty ,5} \right)\]

Answer 1

Hint: The given inequalities contain only one variable \[x\]. To solve the equation having one variable, we need only one equation to solve for that variable. At first, we will simplify the given inequalities to a possible extent by using operations of inequality. We will also use the Least common multiple to add the terms, if needed. We will use any of addition, subtraction, multiplication and division possible to reach the value of \[x\]. At the end we will have two ranges of values of \[x\]. The common values of \[x\] in both the solutions will be the required answer.

Complete step-by-step solution:
We have, \[\dfrac{{4x}}{3} - \dfrac{9}{4} < x + \dfrac{3}{4}\]
Adding \[\dfrac{9}{4}\] both sides, we get,
\[\dfrac{{4x}}{3} - \dfrac{9}{4} + \dfrac{9}{4} < x + \dfrac{3}{4} + \dfrac{9}{4}\]
\[ \Rightarrow \dfrac{{4x}}{3} < x + \dfrac{{12}}{4}\]
Now, Multiplying 3 both sides, we get,
\[4x < 3x + 9\]
Subtract \[3x\] from both sides, we obtain,
\[x < 9\] \[......\left( 1 \right)\]
Also, we have, \[\dfrac{{7x - 1}}{3} - \dfrac{{7x + 2}}{6} > x\]
Multiplying by \[6\] both sides, we get,
\[2\left( {7x - 1} \right) - \left( {7x + 2} \right) > 6x\]
Simplifying it,
\[ 14x - 2 - 7x - 2 > 6x \\
   \Rightarrow 7x - 4 > 6x \]
Add \[4\] both sides,
\[7x > 6x + 4\]
Subtract \[6x\] from both sides, we obtain,
\[x > 4\] \[.......\left( 2 \right)\]
Now, from equation \[\left( 1 \right)\& \left( 2 \right)\], we can say,
\[4 < x < 9\]
B. \[\left( {4,\left. 9 \right)} \right.\]

Note: The difference between an inequality and an equation is the result that we get after solving it. On solving an equation in one variable, we get a unique point on a number line whereas on solving an inequality with one variable, we get a range of points or a region on the number line. Like, in the above solution, we get a range of points for \[x\], \[4 < x < 9\]
We have used the open interval \[\left( { } \right)\] bracket for the above range which means the range doesn’t include \[4,9\].
Closed interval bracket \[\left[ { } \right]\] means it includes all the values in a range including the interval values.
Rules for solving the inequalities: (1). Eliminate fraction and decimals. (2). Flip the inequality symbol if multiplying or dividing by a negative number. (3). Isolate terms containing variables. (4). Combine like terms. (5). Divide by numerical coefficient of variable (only and only if coefficient is greater than 1).\[\]