Math  /  Algebra

QuestionSolve the inequality 4x9<7-4 x-9<7.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice A. The solution set is \square \square. (Type your answer in interval notation. Use integers or fractions for any numbers in the expression.) B. The solution is the empty set.

Studdy Solution

STEP 1

What is this asking? We need to find all the values of xx that make the inequality 4x9<7-4x - 9 < 7 true. Watch out! Remember to flip the inequality sign when multiplying or dividing by a negative number!

STEP 2

1. Isolate the term with xx.
2. Solve for xx.
3. Express the solution in interval notation.

STEP 3

We want to get 4x-4x by itself, so let's add **9** to both sides of the inequality: 4x9+9<7+9-4x - 9 + 9 < 7 + 9 4x<16-4x < 16Why did we add 9?
Because adding 9 to 9-9 results in zero, and adding zero does not change the value of the other terms.

STEP 4

Now, we want to get xx all alone.
Since xx is multiplied by 4-4, we'll divide both sides by 4-4.
Remember, when dividing by a negative number, we *flip* the inequality sign! 4x4>164 \frac{-4x}{-4} > \frac{16}{-4} x>4 x > -4 Dividing 4x-4x by 4-4 results in 1x1 \cdot x, which is just xx.
And 1616 divided by 4-4 is 4-4.

STEP 5

Since xx is greater than 4-4, our solution starts at 4-4 and goes all the way to infinity!
We use a parenthesis for 4-4 because xx is *strictly greater than* 4-4, not equal to it.
Infinity always gets a parenthesis.
So, our interval notation is: (4,) (-4, \infty)

STEP 6

The solution set is (4,)(-4, \infty).
So we choose A, and fill in the box with (4,)(-4, \infty).

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