Answer:
48. [tex]x \geq -7[/tex] or [tex]x \leq 31[/tex]
51. [tex]x \leq 13[/tex] or [tex]x \geq 3[/tex]
54. [tex]x < \frac{5}{2}[/tex] or [tex]x > \frac{-15}{2}[/tex]
57. [tex]x < \frac{18}{7}[/tex] or [tex]x > -4[/tex]
Step-by-step explanation:
48. [tex]| 12 - x | \leq 19[/tex]
-There are two equations:
Equation 1:
[tex]12 - x \leq 19[/tex]
or
Equation 2:
[tex]12 - x \geq -19[/tex]
Solving equation 1:
[tex]12 - x \leq 19[/tex]
[tex]12- 12 - x \leq 19 - 12[/tex]
[tex]-x \leq 7[/tex]
[tex]\frac{-x}{-1} \leq \frac{7}{-1}[/tex]
[tex]x \geq -7[/tex] (Inequality sign changed, because of dividing by a negative number)
-Solving equation 2:
[tex]12 - x \geq -19[/tex]
[tex]12 - 12 - x \geq -19 - 12[/tex]
[tex]-x \geq -31[/tex]
[tex]\frac{-x}{-1} \geq \frac{-31}{-1}[/tex]
[tex]x \leq 31[/tex] (Inequality sign changed, because of dividing by a negative number)
-Answers:
[tex]x \geq -7[/tex] or [tex]x \leq 31[/tex]
51. [tex]| x - 8 | \leq 5[/tex]
-There are two equations:
Equation 1:
[tex]x - 8 \leq 5[/tex]
or
Equation 2:
[tex]x - 8 \geq - 5[/tex]
-Solving equation 1:
[tex]x - 8 \leq 5[/tex]
[tex]x - 8 + 8 \leq 5 + 8[/tex]
[tex]x \leq 13[/tex]
-Solving equation 2:
[tex]x - 8 \geq - 5[/tex]
[tex]x - 8 + 8 \geq - 5 + 8[/tex]
[tex]x \geq 3[/tex]
Answers:
[tex]x \leq 13[/tex] or [tex]x \geq 3[/tex]
54. [tex]| 4x + 10 | < 20[/tex]
-There are two equations:
Equation 1:
[tex]4x + 10 < 20[/tex]
or
Equation 2:
[tex]4x + 10 > -20[/tex]
-Solving equation 1:
[tex]4x + 10 < 20[/tex]
[tex]4x + 10 - 10 < 20 - 10[/tex]
[tex]4x < 10[/tex]
[tex]\frac{4x}{4} < \frac{10}{4}[/tex]
[tex]x < \frac{5}{2}[/tex]
-Solving equation 2:
[tex]4x + 10 > -20[/tex]
[tex]4x + 10 - 10 > -20 - 10[/tex]
[tex]4x > -30[/tex]
[tex]\frac{4x}{4} > \frac{-30}{4}[/tex]
[tex]x > \frac{-15}{2}[/tex]
-Answers:
[tex]x < \frac{5}{2}[/tex] or [tex]x > \frac{-15}{2}[/tex]
57. [tex]| 7x + 5 | < 23[/tex]
-There are two equations:
Equation 1:
[tex]7x + 5 < 23[/tex]
or
Equation 2:
[tex]7x + 5 > -23[/tex]
-Solving equation 1:
[tex]7x + 5 < 23[/tex]
[tex]7x + 5 - 5 < 23 - 5[/tex]
[tex]7x < 18[/tex]
[tex]\frac{7x}{7} < \frac{18}{7}[/tex]
[tex]x < \frac{18}{7}[/tex]
Solving equation 2:
[tex]7x + 5 > -23[/tex]
[tex]7x + 5 - 5 > -23 - 5[/tex]
[tex]7x > -28[/tex]
[tex]\frac{7x}{7} > \frac{-28}{7}[/tex]
[tex]x > -4[/tex]
Answers:
[tex]x < \frac{18}{7}[/tex] or [tex]x > -4[/tex]
And we are finished.
