How do you solve the inequality #abs(3x-5)

Question

How do you solve the inequality #abs(3x-5)<=4# and write your answer in interval notation?

Eva Abramson · Accepted Answer

See a solution process below:

The absolute value function takes any term and transforms it to its non-negative form. Therefore, we must solve the term within the absolute value function for both its negative and positive equivalent. #-4 <= 3x - 5 <= 4# First, add #color(red)(5)# to each segment of the system of inequalities to isolate the #x# term while keeping the system balanced: #-4 + color(red)(5) <= 3x - 5 + color(red)(5) <= 4 + color(red)(5)# #1 <= 3x - 0 <= 9# #1 <= 3x <= 9# Now, divide each segment by #color(red)(3)# to solve for #x# while keeping the system balanced: #1/color(red)(3) <= (3x)/color(red)(3) <= 9/color(red)(3)# #1/3 <= (color(red)(cancel(color(black)(3)))x)/cancel(color(red)(3)) <= 3# #1/3 <= x <= 3# Or #x >= 1/3# and #x <= 3# Or, in interval notation: #[1/3, 3]#

Eva Abramson · Answer

undefined To solve the inequality $ |3x - 5| \leq 4 $ and express the solution in interval notation:

1. Solve for the two cases of the absolute value:
   - $ 3x - 5 \leq 4 $ and $ 3x - 5 \geq -4 $

2. Solve each inequality separately:
   - For $ 3x - 5 \leq 4 $, add 5 to both sides and divide by 3: $ 3x \leq 9 $, so $ x \leq 3 $
   - For $ 3x - 5 \geq -4 $, add 5 to both sides and divide by 3: $ 3x \geq 1 $, so $ x \geq \frac{1}{3} $

3. Combine the solutions:
   - The solution is $ \frac{1}{3} \leq x \leq 3 $

4. Express the solution in interval notation:
   - $ \left[ \frac{1}{3}, 3 \right] $

Jace Anderson · Answer

undefined To solve the inequality $|3x - 5| \leq 4$ and write the answer in interval notation, follow these steps:

1. Split the absolute value inequality into two separate inequalities: $3x - 5 \leq 4$ and $3x - 5 \geq -4$.
2. Solve each inequality separately for $x$.
3. Combine the solutions into one interval.

Solving the first inequality:
$$3x - 5 \leq 4$$
$$3x \leq 9$$
$$x \leq \frac{9}{3}$$
$$x \leq 3$$

Solving the second inequality:
$$3x - 5 \geq -4$$
$$3x \geq 1$$
$$x \geq \frac{1}{3}$$

Combining the solutions:
The solution set for the inequality $|3x - 5| \leq 4$ is the interval $[ \frac{1}{3}, 3 ]$.

How do you solve the inequality #abs(3x-5)&lt;=4# and write your answer in interval notation?

How do you solve the inequality #abs(3x-5)<=4# and write your answer in interval notation?