How do you solve and write the following in interval notation: #|2x-1|+7=1#?

Question

How do you solve and write the following in interval notation: #|2x-1|+7>=1#?

Nathan Axel · Accepted Answer

The solution set for #|2x-1|+7>=1# is #{x in RR}#

Subtract 7 from both sides

#|2x-1| >= -6#

The left hand side is always considered as positive. So it can not be equal to #-6# Thus #|2x-1|+7>=1# is wrong. It should read #|2x-1|+7>1#

The form of #|2x-1|+7>1 # means that all real values of x will make the inequality be true.

The solution set for #|2x-1|+7>=1# is #{x in RR}#

To illustrate this, I shall substitute #1/2# for x:

#|2(1/2)-1|+7 >=1#

#|0|+ 7>=1#

#7 >=1 #

This inequality is true and I have chosen the value for x where any other value will only make the number on the left side become greater. Therefore, the solution set is all real values of x.

Savannah Anderson · Answer

undefined To solve the inequality |2x - 1| + 7 ≥ 1 and write the solution in interval notation, follow these steps:

1. Subtract 7 from both sides of the inequality:
   |2x - 1| + 7 - 7 ≥ 1 - 7
   |2x - 1| ≥ -6

2. Remove the absolute value bars by considering two cases:
   Case 1: 2x - 1 ≥ 0 (when the expression inside the absolute value is non-negative)
   Case 2: 2x - 1 < 0 (when the expression inside the absolute value is negative)

3. For Case 1 (2x - 1 ≥ 0):
   2x - 1 ≥ 0
   2x ≥ 1
   x ≥ 1/2

4. For Case 2 (2x - 1 < 0):
   2x - 1 < 0
   2x < 1
   x < 1/2

5. Combine the solutions from both cases:
   x ≥ 1/2 or x < 1/2

6. Write the solution in interval notation:
   x ∈ (-∞, 1/2] ∪ (1/2, ∞)

This notation represents all real numbers less than or equal to 1/2 combined with all real numbers greater than 1/2, excluding 1/2 itself.

Kennedy Adams · Answer

undefined To solve the inequality $|2x - 1| + 7 \geq 1$, we first isolate the absolute value term:

$$ |2x - 1| \geq 1 - 7 = -6 $$

Absolute value cannot be negative, so the inequality $|2x - 1| \geq -6$ holds true for all real numbers $x$. Therefore, we don't need to consider the absolute value.

We are left with the inequality:

$$ 7 \geq 1 $$

This inequality is always true.

Thus, the solution to the inequality $|2x - 1| + 7 \geq 1$ is all real numbers, or in interval notation: $[-∞, +∞]$.

How do you solve and write the following in interval notation: #|2x-1|+7&gt;=1#?

How do you solve and write the following in interval notation: #|2x-1|+7>=1#?