Chapter 5: Linear Inequalities

Ans: $2 \leqslant 3x - 4 \leqslant 5$

$\Rightarrow 2 + 4 \leqslant 3x - 4 + 4 \leqslant 5 + 4$

$\Rightarrow 6 \leqslant 3x \leqslant 9$

$\Rightarrow 2 \leqslant x \leqslant 3$

As a result, the solutions of the following inequality are all real values higher than or equal to 2 but less than or equal to 3. For the given inequality, the solution set is [2,3].

Ans: $6 \leqslant  - 3(2x - 4) < 12$

$\Rightarrow 2 \leqslant  - (2x - 4) < 4$

$\Rightarrow  - 2 \geqslant 2x - 4 >  - 4$

$\Rightarrow 4 - 2 \geqslant 2x > 4 - 4$

$\Rightarrow 2 \geqslant 2x > 0$

$\Rightarrow 1 \geqslant x > 0$

As a result, the set of solutions for the given inequality is [1 ,0 ).

Ans: $ - 3 \leqslant 4 - \dfrac{{7x}}{2} \leqslant 18$

$\Rightarrow  - 3 - 4 \leqslant  - \dfrac{{7x}}{2} \leqslant 18 - 4$

$\Rightarrow  - 7 \leqslant  - \dfrac{{7x}}{2} \leqslant 14$

$\Rightarrow 7 \geqslant \dfrac{{7x}}{2} \geqslant  - 14$

$\Rightarrow 1 \geqslant \dfrac{x}{2} \geqslant  - 2$

$\Rightarrow 2 \geqslant x \geqslant  - 4$

As a result, the set of solutions for the given inequality is $[ - 4,2]$.

Ans: $ - 15 < \dfrac{{3(x - 2)}}{5} \leqslant 0$

$\Rightarrow  - 75 < 3(x - 2) \leqslant 0$

$\Rightarrow  - 25 < x - 2 \leqslant 0$

$\Rightarrow  - 25 + 2 < x \leqslant 2$

$\Rightarrow  - 23 < x \leqslant 2$

As a result, the set of solutions for the given inequality is (-23, 2]

Ans: \[ - 12 < 4 - \dfrac{{3x}}{{ - 5}} \leqslant 2\]

$\Rightarrow  - 12 - 4 < \dfrac{{ - 3x}}{{ - 5}} \leqslant 2 - 4$

$\Rightarrow  - 16 < \dfrac{{3x}}{5} \leqslant  - 2$

$\Rightarrow  - 80 < 3x \leqslant  - 10$

$\Rightarrow \dfrac{{ - 80}}{3} < x \leqslant \dfrac{{ - 10}}{3}$

As a result, the set of solutions for the given inequality is $\left( {\dfrac{{ - 80}}{3},\dfrac{{ - 10}}{3}} \right]$.

Ans: $7 \leqslant \dfrac{{(3x + 11)}}{2} \leqslant 11$

$\Rightarrow 14 \leqslant 3x + 11 \leqslant 22$

$\Rightarrow 14 - 11 \leqslant 3x \leqslant 22 - 11$

$\Rightarrow 3 \leqslant 3x \leqslant 11$

$\Rightarrow 1 \leqslant x \leqslant \dfrac{{11}}{3}$

As a result, the set of solutions for the given inequality is $\left[ {1,\dfrac{{11}}{3}} \right]$.

Ans: $5x + 1 >  - 24 \Rightarrow 5x >  - 25$

$\Rightarrow x >  - 5 \ldots .(1)$

$5x - 1 < 24 \Rightarrow 5x < 25$

$\Rightarrow x < 5$

From (1) and ( 2 ), The solution set for the given system of inequalities can be deduced to be$( - 5,5)$. On a number line, the solution to the above system of inequalities can be expressed as

Ans: $2(x - 1) < x + 5 \Rightarrow 2x - 2 < x + 5 \Rightarrow 2x - x < 5 + 2$

$\Rightarrow x < 7$

(1) $3(x + 2) > 2 - x \Rightarrow 3x + 6 > 2 - x \Rightarrow 3x + x > 2 - 6$

$\Rightarrow 4x >  - 4$

$\Rightarrow x >  - 1 \ldots  \ldots (2)$

From (1) and (2), The solution set for the given system of inequalities can be deduced to be $( - 1,7)$. On a number line, the solution to the above system of inequalities can be expressed as

Ans: $3x - 7 > 2(x - 6) \Rightarrow 3x - 7 > 2x - 12 \Rightarrow 3x - 2x >  - 12 + 7$

$\Rightarrow x >  - 5 \ldots  \ldots  \ldots .(1)$

$ 6 - x > 11 - 2x \Rightarrow  - x + 2x > 11 - 6$

$\Rightarrow x > 5$

From (1) and (2), The solution set for the given system of inequalities can be deduced to be

$(5,\infty )$. On a number line, the solution to the above system of inequalities can be expressed as

Ans: $5(2x - 7) - 3(2x + 3) \leqslant 0 \Rightarrow 10x - 35 - 6x - 9 \leqslant 0 \Rightarrow 4x - 44 \leqslant 0 \Rightarrow 4x \leqslant 44$

$\Rightarrow x \leqslant 11$

$2x + 19 \leqslant 6x + 47 \Rightarrow 19 - 47 \leqslant 6x - 2x \Rightarrow  - 28 \leqslant 4x$

$\Rightarrow  - 7 \leqslant x$

From (1) and (2), The solution set for the given system of inequalities can be deduced to be $[ - 7,11]$. On a number line, the solution to the above system of inequalities can be expressed as