Question - Solving Inequalities Involving Absolute Value
mathsolving absolute value inequalitiesdistance on number lineinequalities involving absolute valueabsolute value inequality cases
Solution:
This is an inequality involving the absolute value of a variable $$ d $$. To solve for $$ d $$, we need to consider two cases because the absolute value of a number represents its distance from 0 on the number line, regardless of direction.The inequality is:\[ 3|d| + 5 < 47 \]First, we will isolate the absolute value on one side by subtracting 5 from both sides:\[ 3|d| < 42 \]Next, we will divide both sides by 3 to solve for the absolute value of $$ d $$:\[ |d| < 14 \]Now let's consider the two cases for $$ d $$:1. If $$ d $$ is non-negative ($$ d \geq 0 $$), the absolute value sign can be removed without changing the sign of $$ d $$:\[ d < 14 \]2. If $$ d $$ is negative ($$ d < 0 $$), we remove the absolute value sign and reverse the inequality sign because the absolute value of a negative number is positive:\[ -d < 14 \]\[ d > -14 \]So the solution for the inequality $$ 3|d| + 5 < 47 $$ is:\[ -14 < d < 14 \]This means that $$ d $$ must be greater than -14 and less than 14.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com