Question

Help!! Solving absolute value inequalities

Answer 1

Answer:

47. $x < 7$  or $x > -9$

50. $x > 7$ or $x < 17$

53. $x < \frac{-4}{3}$ or $x > \frac{32}{3}$

56. $x > -28$ or $x < -52$

Step-by-step explanation:

47. $| x + 1| < 8$

-There are two equations:

Equation 1:

$x + 1 < 8$

    or

Equation 2:

$x + 1 > -8$

-Solving equation 1:

$x + 1 < 8$

$x + 1 - 1< 8 - 1$

$x < 7$

-Solving equation 2:

$x + 1 > -8$

$x + 1 - 1 > -8 - 1$

$x > -9$

Answers:

$x < 7$  or $x > -9$

50. $| x + 5 | > 12$

-There are two equations:

Equation 1:

$x + 5 > 12$

     or

Equation 2:

$x + 5 < -12$

-Solving equation 1:

$x + 5 > 12$

$x + 5 - 5 > 12 - 5$

$x > 7$

-Solving equation 2:

$x + 5 < -12$

$x + 5 - 5 < -12 - 5$

$x < -17$

-Answers:

$x > 7$ or $x < -17$

53. $| 14 - 3x | > 18$

-There are two equations:

Equation 1:

$14 - 3x > 18$

      or

Equations 2:

$14 - 3x < -18$

-Solving equation 1:

$14 - 3x > 18$

$14 - 14 - 3x > 18 -14$

$-3x > 4$

$\frac{-3x}{-3} > \frac{4}{-3}$

$x < \frac{-4}{3}$ (Inequality sign changed, because of dividing by a negative number)

-Solve equation 2:

$14 - 3x < -18$

$14 - 14 - 3x < -18 -14$

$-3x < -32$

$\frac{-3x}{-3} < \frac{-32}{-3}$

$x > \frac{32}{3}$ (Inequality sign changed, because of dividing by a negative number)

-Answers:

$x < \frac{-4}{3}$ or $x > \frac{32}{3}$

56. $| 20 + \frac{1}{2}x | > 6$

-Switch the equation:

$| \frac{1}{2}x + 20 | > 6$

-There are two equations:

Equations 1:

$\frac{1}{2}x + 20 > 6$

     or

Equation 2:

$\frac{1}{2}x + 20 < -6$

-Solving equation 1:

$\frac{1}{2}x + 20 > 6$

$\frac{1}{2}x + 20 - 20 > 6 - 20$

$\frac{1}{2}x > -14$

$2 \times \frac{1}{2}x > 2 \times -14$

$x > -28$

-Solving equation 2:

$\frac{1}{2}x + 20 < -6$

$\frac{1}{2}x + 20 - 20 < -6 - 20$

$\frac{1}{2}x < -26$

$2 \times \frac{1}{2}x < 2 \times -26$

$x < -52$

-Answers:

$x > -28$ or $x < -52$

And were done.