How do you solve and write the following in interval notation: #7 ≥ 2x − 5# OR #(3x − 2) / 44#?

Question

How do you solve and write the following in interval notation: #7 ≥ 2x − 5# OR #(3x − 2) / 4>4#?

Kennedy Adams · Accepted Answer

#(-oo, oo)#

First, solve each inequality. I'll solve the first one first.

#7 >= 2x-5# #12 >= 2x# #6 >= x#

Therefore, x could be any number less than or equal to 6. In interval notation, this looks like:

#(-oo, 6]#

The parenthesis means that the lower end is not a solution, but every number above it is. (In this case, the lower end is infinity, so a parenthesis must be used, since infinity is not a real number and so it cannot be a solution.) The bracket means that the upper end is a solution. In this case, it indicates that not only could #x# be any number less than 6, but it could also be 6.

Let's try the second example:

#(3x-2)/4 > 4#

#3x-2 > 16# #3x > 18# #x > 6#

Therefore, x could be any number greater than 6, but x couldn't be 6, since that would make the two sides of the inequality equal. In interval notation, this looks like:

#(6, oo)#

The parentheses mean that neither end of this range is included in the solution set. In this case, it indicates that neither 6 nor infinity are solutions, but every number in between 6 and infinity is a solution (that is, every real number greater than 6 is a solution).

Now, the problem used the word "OR", meaning that either of these equations could be true. That means that either #x# is on the interval #(-oo, 6]# or the interval #(6, oo)#. In other words, #x# is either less than or equal to 6, or it is greater than 6. When you combine these two statements, it becomes clear that #x# could be any real number, since no matter what number #x# is, it will fall in one of these intervals. The interval "all real numbers" is written like this:

#(-oo, oo)#

Final Answer

Avery Alexander · Answer

undefined To solve the compound inequality $7 \geq 2x - 5$ or $\frac{3x - 2}{4} > 4$, we solve each inequality separately and then combine the results.

For the first inequality:
$$7 \geq 2x - 5$$
$$7 + 5 \geq 2x$$
$$12 \geq 2x$$
$$6 \geq x$$

For the second inequality:
$$\frac{3x - 2}{4} > 4$$
$$3x - 2 > 16$$
$$3x > 18$$
$$x > 6$$

Combining the solutions, we have $x \leq 6$ or $x > 6$. This can be expressed in interval notation as:
$(- \infty, 6] \cup (6, \infty)$.

How do you solve and write the following in interval notation: #7 ≥ 2x − 5# OR #(3x − 2) / 4&gt;4#?

How do you solve and write the following in interval notation: #7 ≥ 2x − 5# OR #(3x − 2) / 4>4#?