a/24
[tex] - 1 < \frac{3x + 7}{2} \leqslant 11 \\ - 1 < 3x + 7 \leqslant 11 | \times 2 \\ - 2 < 3x \leqslant 22 | - 7 \\ - 9 < x \leqslant 15 | \div 3 \\ - 3 < x \leqslant 5 [/tex]
A= (-3,5]
[tex] - 2 \leqslant \frac{5x + 9}{8} < 3 \\ - 2 \leqslant 5x + 9 < 3 | \times 8 \\ - 16 \leqslant 5x < 24 | - 9 \\ - 25 \leqslant x < 15 | \div 5 \\ - 5 \leqslant x < 3[/tex]
B= [-5,3)
AUB= [-5,5]
A⋂B= (-3,3)
b/24
[tex] - 1 < \frac{4x + 6}{2} < 9 \\ - 1 < 4x + 6 < 9 | \times 2 \\ - 2 < 4x < 18 | - 6 \\ - 8 < x < 12 | \div 4 \\ - 2 < x < 3[/tex]
A= (-2,3)
[tex] - 7 < \frac{6x + 10}{2} < 11 \\ - 7 < 6x + 10 < 11 | \times 2 \\ - 14 < 6x < 22 | - 10 \\ - 24 < x < 12 | \div 6 \\ - 4 < x < 2[/tex]
B= (-4,2)
AUB= (-4,3)
A⋂B= (-2,2)
c/24
[tex] - 9 < \frac{5x + 12}{2} < 16 \\ - 9 < 5x + 12 < 16| \times 2 \\ - 18 < 5x < 32 | - 12 \\ - 30 < x < 20 | \div 5 \\ - 6 < x < 4[/tex]
A= (-6,4)
[tex] - 8 < \frac{7x + 12}{2} \leqslant 27 \\ - 8 < 7x + 12 \leqslant 27 | \times 2 \\ - 16 < 7x \leqslant 54 | - 12 \\ - 28 < x \leqslant 42 | \div 7 \\ - 4 < x \leqslant 6[/tex]
B= (-4,6]
AUB= (-6,6]
A⋂B= (-4,4)
d/24
[tex] - 1 < \frac{4x + 10}{2} < 13 \\ - 1 < 4x + 10 < 13 | \times 2 \\ - 2 < 4x < 26 | - 10 \\ - 12 < x < 16 | \div 4 \\ - 3 < x < 4[/tex]
A= (-3,4)
[tex]3 < \frac{2x + 14}{2} < 12 \\ 3 < 2x + 14 < 12 | \times 2 \\ 6 < 2x < 24 | - 14 \\ - 8 < x < 10 | \div 2 \\ - 4 < x < 5[/tex]
B= (-4,5)
AUB= (-4,5)
A⋂B= (-3,4)
25.a)
|2x-7| ≤ 3 => -3≤ 2x-7 ≤ 3 | +7 => 4≤ 2x ≤ 10 | ÷2 => 2≤ x≤ 5
A= [2,5]
|2x+1/9≤ 1 => -1≤ 2x+1/9 ≤1 |×9 => -9≤ 2×+1≤ 9 |-1
=> -10≤ 2x≤ 8 |÷2 => -5≤ x ≤ 4
B= [-5,4]
AUB= [-5,5]
A⋂B= [2,4]
b) |x+2| < 5 => -5< x+2< 5 |-2 => -7< x < 3
A= (-7,3)
|x-3| < 4 => -4< x-3 <4 |+3 => -1< x < 7
B= (-1,7)
AUB= (-7,7)
A⋂B= (-1,3)
c) |2x-3|+3≤ 14 => -14≤ 2x-3+3 ≤ 14 => -14≤ 2x≤ 14 |÷2
=> -7≤ x ≤ 7
A= [-7,7]
-1< 3x+7/8 < 2 |×8=> -8< 3x+7 < 16 |-7 => -15< 3x < 9 |÷3
=> -5< x < 3
B= (-5,3)
AUB= [-7,7]
A⋂B= (-5,3)
d) |2x-7| +5≤ 8 => -8≤ 2x-2≤ 8 |+2 => -6≤ 2x ≤ 10 |÷2
=> -3≤ x ≤ 5
A= [-3,5]
|4x-3/3| ≤ 3 => -3 ≤ 4x-3 ≤ 3 |×3 => -9≤ 4x ≤ 9 |+3
=> -6≤ x ≤ 12 |÷4 => -1,5 ≤ x ≤ 3
B= [-1,5; 3]
AUB= [-3,5]
A⋂B= [-1,5;3]
26/a)
|2x-1| ≤ 11 => -11≤ 2x-1 ≤ 11 |+1 => -10≤ 2x ≤ 12 |÷2
=> -5≤ x ≤ 6
A= [-5,6]
|2x+1| >7 => -7> 2x+1 >7 |-1 => -8> 2x > 6 |÷2 => -4> x > 3
B= (-4,3)
AUB= [-5,6]
A⋂B= (-4,3)
b) |2x-1| >3 => -3 > 2x-1 >3 |+1 => -2> 2x> 4 |÷2 => -1 > x > 2
A= (-1,2)
|2x+1| < 9 => -9< 2x+1 <9 |-1 => -10< 2x< 8 |÷2 => -5< x < 4
B= (-5,4)
AUB= (-5,4)
A⋂B= (-1,2)
c) |3x+7/5| ≤ 4 => -4≤ 3x+7 ≤ 4 |×5 => -20≤ 3x ≤ 20 |-7
=> -27≤ x ≤ 13 |÷3 => -9≤ x ≤ 4.3
A= [-9,4.3]
|2x-5| ≥ 3 => -3≥ 2x ≥3 |+5 => 2≥ x ≥ 8 |÷2 => 1 ≥ x ≥ 4
B= [1,4]
AUB= [-9, 4.3]
A⋂B= [1,4]
d) |x-3| > 2 => -2> x-3 >2 |+3 => 1> x > 5
A= (1,5)
1≤ 3x+7/4 ≤ 7 |×4 => 4≤ 3x+7 ≤ 28 |-7
=> -3≤ 3x ≤ 21 |÷3 => -1≤ x ≤ 7
B= [-1, 7]
AUB= [-1,7]
A⋂B= (1,5)
