How do you solve #abs(2x-3)

Question

How do you solve #abs(2x-3)<=3#?

Kennedy Adams · Accepted Answer

#0<=x<=3# #"inequalities of the type "|x|<=a# #"always have solutions of the form"# #-a<=x<=a# #rArr-3<=2x-3<=3# #"add 3 to each interval"# #cancel(-3)cancel(+3)<=2xcancel(-3)cancel(+3)<=3+3# #rArr0<=2x<=6# #"divide each interval by 2"# #rArr0<=x<=3" is the solution"# #x in[0,3]larrcolor(blue)"in interval notation"#

Kennedy Adams · Answer

undefined To solve the inequality $ |2x - 3| \leq 3 $, follow these steps:

1. Set up two inequalities:
   $$ 2x - 3 \leq 3 $$
   $$ -(2x - 3) \leq 3 $$

2. Solve each inequality separately:
   For the first inequality:
   $$ 2x - 3 \leq 3 $$
   Add 3 to both sides:
   $$ 2x \leq 6 $$
   Divide both sides by 2:
   $$ x \leq 3 $$

For the second inequality:
   $$ -(2x - 3) \leq 3 $$
   Distribute the negative sign:
   $$ -2x + 3 \leq 3 $$
   Subtract 3 from both sides:
   $$ -2x \leq 0 $$
   Divide both sides by -2 (note that when dividing by a negative number, the inequality sign flips):
   $$ x \geq 0 $$

3. Combine the solutions:
   The solution to the inequality $ |2x - 3| \leq 3 $ is the intersection of the solutions to the two inequalities:
   $$ 0 \leq x \leq 3 $$

Julia Adams · Answer

undefined To solve the inequality $ |2x - 3| \leq 3 $, you need to consider two cases: when $ 2x - 3 $ is positive or zero, and when $ 2x - 3 $ is negative.

1. When $ 2x - 3 \geq 0 $:
   In this case, the inequality becomes $ 2x - 3 \leq 3 $.
   Solve for $ x $: 
   $$ 2x - 3 \leq 3 $$
   $$ 2x \leq 6 $$
   $$ x \leq 3 $$

2. When $ 2x - 3 < 0 $:
   In this case, we need to consider the opposite of the expression inside the absolute value.
   $$ -(2x - 3) \leq 3 $$
   $$ -2x + 3 \leq 3 $$
   $$ -2x \leq 0 $$
   $$ x \geq 0 $$

So, combining both cases, the solution to the inequality $ |2x - 3| \leq 3 $ is $ x \in [0, 3] $.

How do you solve #abs(2x-3)&lt;=3#?

How do you solve #abs(2x-3)<=3#?