0.08e: Exercises- Linear Inequalities

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Jun 22, 2022
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- 0.08: Review - Linear Inequalities in One Variable
- 0.09: Review - Graphs of Linear Equations

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A: Check a solution

Exercise $\PageIndex{1}$

$\bigstar$ Determine whether or not the given value is a solution.

$5 x - 1 < - 2 ; x = - 1$
$- 3 x + 1 > - 10 ; x = 1$
$2 x - 3 < - 5 ; x = 1$
$5 x - 7 < 0 ; x = 2$

$9 y - 4 \geq 5 ; y = 1$
$- 6 y + 1 \leq 3 ; y = - 1$
$12 a + 3 \leq - 2 ; a = - \dfrac { 1 } { 3 }$

$25 a - 2 \leq - 22 ; a = - \dfrac { 4 } { 5 }$
$- 10 < 2 x - 5 < - 5 ; x = - \dfrac { 1 } { 2 }$
$3 x + 8 < - 2 \text { or } 4 x - 2 > 5 ; x = 2$

Answers to odd exercises:

1. Yes

3. No

5. Yes

7. No

9. Yes

B: Solve Linear Inequalities

Exercise $\PageIndex{2}$

$\bigstar$ Graph all solutions on a number line and provide the corresponding interval notation.

$3 x + 5 > - 4$
$2 x + 1 > - 1$
$5 - 6 y < - 1$
$7 - 9 y > 43$
$6 - a \leq 6$
$- 2 a + 5 > 5$
$\dfrac { 5 x + 6 } { 3 } \leq 7$
$\dfrac { 4 x + 11 } { 6 } \leq \dfrac { 1 } { 2 }$
$\dfrac { 1 } { 2 } y + \dfrac { 5 } { 4 } \geq \dfrac { 1 } { 4 }$
$\dfrac { 1 } { 12 } y + \dfrac { 2 } { 3 } \leq \dfrac { 5 } { 6 }$
$2 ( 3 x + 14 ) < - 2$

$5 ( 2 y + 9 ) > - 15$
$5 - 2 ( 4 + 3 y ) \leq 45$
$- 12 + 5 ( 5 - 2 x ) < 83$
$6 ( 7 - 2 a ) + 6 a \leq 12$
$2 a + 10 ( 4 - a ) \geq 8$
$9 ( 2 t - 3 ) - 3 ( 3 t + 2 ) < 30$
$- 3 ( t - 3 ) - ( 4 - t ) > 1$
$\dfrac { 1 } { 2 } ( 5 x + 4 ) + \dfrac { 5 } { 6 } x > - \dfrac { 4 } { 3 }$
$\dfrac { 2 } { 5 } + \dfrac { 1 } { 6 } ( 2 x - 3 ) \geq \dfrac { 1 } { 15 }$
$5 x - 2 ( x - 3 ) < 3 ( 2 x - 1 )$
$3 ( 2 x - 1 ) - 10 > 4 ( 3 x - 2 ) - 5 x$

$- 3 y \geq 3 ( y + 8 ) + 6 ( y - 1 )$
$12 \leq 4 ( y - 1 ) + 2 ( 2 y + 1 )$
$- 2 ( 5 t - 3 ) - 4 > 5 ( - 2 t + 3 )$
$- 7 ( 3 t - 4 ) > 2 ( 3 - 10 t ) - t$
$\dfrac { 1 } { 2 } ( x + 5 ) - \dfrac { 1 } { 3 } ( 2 x + 3 ) > \dfrac { 7 } { 6 } x + \dfrac { 3 } { 2 }$
$- \dfrac { 1 } { 3 } ( 2 x - 3 ) + \dfrac { 1 } { 4 } ( x - 6 ) \geq \dfrac { 1 } { 12 } x - \dfrac { 5 } { 4 }$
$4 ( 3 x + 4 ) \geq 3 ( 6 x + 5 ) - 6 x$
$1 - 4 ( 3 x + 7 ) < - 3 ( x + 9 ) - 9 x$
$6 - 3 ( 2 a - 1 ) \leq 4 ( 3 - a ) + 1$
$12 - 5 ( 2 a + 6 ) \geq 2 ( 5 - 4 a ) - a$

Answers to odd exercises:

11. $( - 3 , \infty )$ ;

$a2d07ff66453c86a51eb9ee5ca94a731.png$

Figure 1.8.11

13. $( 1 , \infty )$ ;

15. $[ 0 , \infty )$ ;

17. $( - \infty , 3 ]$ ;

19. $[ - 2 , \infty )$ ;

21. $( - \infty , - 5 )$ ;

23. $[ - 8 , \infty )$ ;

25. $[ 5 , \infty )$ ;

27. $( - \infty , 7 )$ ;

29. $( - 1 , \infty )$ ;

31. $( 3 , \infty )$ ;

33. $\left( - \infty , - \dfrac { 3 } { 2 } \right]$ ;

35. $\emptyset$ ;

37. $( - \infty , 0 )$ ;

39. $\mathbb { R }$ ;

41. $[ - 2 , \infty )$ ;

C: Solve Compound Linear Inequalities

Exercise $\PageIndex{3}$

$\bigstar$ Graph all solutions on a number line and provide the corresponding interval notation.

$- 1 < 2 x + 1 < 9$
$- 4 < 5 x + 11 < 16$
$- 7 \leq 6 y - 7 \leq 17$
$- 7 \leq 3 y + 5 \leq 2$
$- 7 < \dfrac { 3 x + 1 } { 2 } \leq 8$
$- 1 \leq \dfrac { 2 x + 7 } { 3 } < 1$
$- 4 \leq 11 - 5 t < 31$
$15 < 12 - t \leq 16$
$- \dfrac { 1 } { 3 } \leq \dfrac { 1 } { 6 } a + \dfrac { 1 } { 3 } \leq \dfrac { 1 } { 2 }$
$- \dfrac { 1 } { 6 } < \dfrac { 1 } { 3 } a + \dfrac { 5 } { 6 } < \dfrac { 3 } { 2 }$
$5 x + 2 < - 3 \text { or } 7 x - 6 > 15$
$4 x + 15 \leq - 1 \text { or } 3 x - 8 \geq - 11$
$8 x - 3 \leq 1 \text { or } 6 x - 7 \geq 8$
$6 x + 1 < - 3 \text { or } 9 x - 20 > - 5$

$8 x - 7 < 1 \text { or } 4 x + 11 > 3$
$10 x - 21 < 9 \text { or } 7 x + 9 \geq 30$
$7 + 2 y < 5 \text { or } 20 - 3 y > 5$
$5 - y < 5 \text { or } 7 - 8 y \leq 23$
$15 + 2 x < - 15 \text { or } 10 - 3 x > 40$
$10 - \dfrac { 1 } { 3 } x \leq 5 \text { or } 5 - \dfrac { 1 } { 2 } x \leq 15$
$9 - 2 x \leq 15 \text { and } 5 x - 3 \leq 7$
$5 - 4 x > 1 \text { and } 15 + 2 x \geq 5$
$7 y - 18 < 17 \text { and } 2 y - 15 < 25$
$13 y + 20 \geq 7 \text { and } 8 + 15 y > 8$
$5 - 4 x \leq 9 \text { and } 3 x + 13 \leq 1$
$17 - 5 x \geq 7 \text { and } 4 x - 7 > 1$
$9 y + 20 \leq 2 \text { and } 7 y + 15 \geq 1$

$21 - 6 y \leq 3 \text { and } - 7 + 2 y \leq - 1$
$- 21 < 6 ( x - 3 ) < - 9$
$0 \leq 2 ( 2 x + 5 ) < 8$
$- 15 \leq 5 + 4 ( 2 y - 3 ) < 17$
$5 < 8 - 3 ( 3 - 2 y ) \leq 29$
$5 < 5 - 3 ( 4 + t ) < 17$
$- 3 \leq 3 - 2 ( 5 + 2 t ) \leq 21$
$- 40 < 2 ( x + 5 ) - ( 5 - x ) \leq - 10$
$- 60 \leq 5 ( x - 4 ) - 2 ( x + 5 ) \leq 15$
$- \dfrac { 1 } { 2 } < \dfrac { 1 } { 30 } ( x - 10 ) < \dfrac { 1 } { 3 }$
$- \dfrac { 1 } { 5 } \leq \dfrac { 1 } { 15 } ( x - 7 ) \leq \dfrac { 1 } { 3 }$
$- 1 \leq \dfrac { a + 2 ( a - 2 ) } { 5 } \leq 0$
$0 < \dfrac { 5 + 2 ( a - 1 ) } { 6 } < 2$