QuestionGraph the solution to the inequality on the number line. $|-3-2 w|<11$

Studdy Solution

STEP 1

What is this asking? We need to find all the values of $w$ that make the absolute value of $-3 - 2w$ less than 11, and then graph those values on a number line. Watch out! Remember that absolute value means "distance from zero," so we're looking for values of $w$ that make $-3 - 2w$ close to zero, within a distance of 11.
Don't forget to flip the inequality sign when multiplying or dividing by a negative number!

STEP 2

1. Rewrite the inequality.
2. Isolate the absolute value.
3. Solve the compound inequality.
4. Graph the solution.

STEP 3

We're given $|-3 - 2w| < 11$ .
Let's rewrite $-3 - 2w$ as $-(3 + 2w)$ , so our inequality becomes $|-(3 + 2w)| < 11$ .

STEP 4

Since the absolute value of a negative number is the same as the absolute value of the positive version of that number, we can rewrite $|-(3 + 2w)|$ as $|3 + 2w|$ .
So, our inequality simplifies to $|3 + 2w| < 11$ .
Much nicer!

STEP 5

Our absolute value expression, $|3 + 2w|$ , is already isolated on one side of the inequality, so we're good to go!
On to the next step!

STEP 6

Since $|3 + 2w| < 11$ , this means that $3 + 2w$ must be between -11 and 11.
We can write this as a compound inequality: $-11 < 3 + 2w < 11$

STEP 7

To isolate the term with $w$ , we'll subtract 3 from all parts of the inequality: $-11 - 3 < 3 + 2w - 3 < 11 - 3$ $-14 < 2w < 8$

STEP 8

Now, we'll divide all parts of the inequality by 2 to solve for $w$ : $\frac{-14}{2} < \frac{2w}{2} < \frac{8}{2}$ $-7 < w < 4$

STEP 9

Let's draw a number line and mark the critical values, -7 and 4.

STEP 10

Since our inequality is $-7 < w < 4$ , we use open circles at -7 and 4 because $w$ cannot equal -7 or 4.
Then, we shade the region between -7 and 4 to represent all the values of $w$ that satisfy the inequality.

STEP 11

The solution to the inequality $|-3 - 2w| < 11$ is $-7 < w < 4$ .
This is represented on the number line with open circles at -7 and 4, and the region between them shaded.

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