Question

Solve the following absolute value inequality.

4x+9
<8
5
x < [?]
x> [ ]
Enter

Answer 1

Answer: x<1

x>-19

Explanation:

`\displaystyle\\(4|x+9|)/(5) < 8`

Expand the modulus and we get a set of inequalities:

`\displaystyle\\\left [ {{(4(x+9))/(5) < 8\ \ \ \ \ (1) } \atop {(4(-(x+9)))/(5) < 8\ \ (2)}} \right. \\\\`

Multiply both parts of inequalities (1) and (2) by 5:

`\displaystyle\\\left [ {{4(x)+4(9) < 8(5)} \atop {4(-x-9) < 5(8)}} \right. \\\\\left [ {{4x+36 < 40} \atop {-4x-36 < 40}} \right. \\\\\left [ {{4x+36-36 < 40-36} \atop {-4x-36+36 < 40+36}} \right. \\\\\left [ {{4x < 4}\ \ \ \ \ \ \ (3) \atop {-4x < 76}\ \ \ \ (4)} \right.`

Divide both parts of inequality (3) by 4

and both parts of inequality (4) by -4:

`\displaystyle\\\left \{ {{x < 1} \atop {x > -19}} \right.`

Thus, x∈(-19,1)