Question

Help!! Solving absolute value inequalities

Answer 1

Answer:

48. $x \geq -7$ or $x \leq 31$

51. $x \leq 13$ or $x \geq 3$

54. $x < \frac{5}{2}$ or $x > \frac{-15}{2}$

57. $x < \frac{18}{7}$ or $x > -4$

Step-by-step explanation:

48. $| 12 - x | \leq 19$

-There are two equations:

Equation 1:

$12 - x \leq 19$

      or

Equation 2:

$12 - x \geq -19$

Solving equation 1:

$12 - x \leq 19$

$12- 12 - x \leq 19 - 12$

$-x \leq 7$

$\frac{-x}{-1} \leq \frac{7}{-1}$

$x \geq -7$ (Inequality sign changed, because of dividing by a negative number)

-Solving equation 2:

$12 - x \geq -19$

$12 - 12 - x \geq -19 - 12$

$-x \geq -31$

$\frac{-x}{-1} \geq \frac{-31}{-1}$

$x \leq 31$ (Inequality sign changed, because of dividing by a negative number)

-Answers:

$x \geq -7$ or $x \leq 31$

51. $| x - 8 | \leq 5$

-There are two equations:

Equation 1:

$x - 8 \leq 5$

    or

Equation 2:

$x - 8 \geq - 5$

-Solving equation 1:

$x - 8 \leq 5$

$x - 8 + 8 \leq 5 + 8$

$x \leq 13$

-Solving equation 2:

$x - 8 \geq - 5$

$x - 8 + 8 \geq - 5 + 8$

$x \geq 3$

Answers:

$x \leq 13$ or $x \geq 3$

54. $| 4x + 10 | < 20$

-There are two equations:

Equation 1:

$4x + 10 < 20$

     or

Equation 2:

$4x + 10 > -20$

-Solving equation 1:

$4x + 10 < 20$

$4x + 10 - 10 < 20 - 10$

$4x < 10$

$\frac{4x}{4} < \frac{10}{4}$

$x < \frac{5}{2}$

-Solving equation 2:

$4x + 10 > -20$

$4x + 10 - 10 > -20 - 10$

$4x > -30$

$\frac{4x}{4} > \frac{-30}{4}$

$x > \frac{-15}{2}$

-Answers:

$x < \frac{5}{2}$ or $x > \frac{-15}{2}$

57. $| 7x + 5 | < 23$

-There are two equations:

Equation 1:

$7x + 5 < 23$

     or

Equation 2:

$7x + 5 > -23$

-Solving equation 1:

$7x + 5 < 23$

$7x + 5 - 5 < 23 - 5$

$7x < 18$

$\frac{7x}{7} < \frac{18}{7}$

$x < \frac{18}{7}$

Solving equation 2:

$7x + 5 > -23$

$7x + 5 - 5 > -23 - 5$

$7x > -28$

$\frac{7x}{7} > \frac{-28}{7}$

$x > -4$

Answers:

$x < \frac{18}{7}$ or $x > -4$

And we are finished.

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