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Math  /  Algebra

QuestionSolve the inequality. Graph the solution. 20≥−3.2(c−4.3)20 \geq-3.2(c-4.3)
The solution is □\square .

Studdy Solution

STEP 1

What is this asking? We need to find all the possible values of cc that make the inequality 20≥−3.2(c−4.3)20 \geq -3.2(c - 4.3) true, and then draw a picture of those values on a number line. Watch out! Remember to flip the inequality sign when multiplying or dividing by a negative number!

STEP 2

1. Isolate the term with cc.
2. Solve for cc.
3. Graph the solution.

STEP 3

First, we'll **distribute** the −3.2-3.2 to both terms inside the parentheses.
This means we multiply −3.2-3.2 by both cc and −4.3-4.3.
We do this to get cc out of the parentheses! −3.2(c−4.3)=−3.2⋅c+(−3.2)⋅(−4.3)=−3.2c+13.76-3.2(c - 4.3) = -3.2 \cdot c + (-3.2) \cdot (-4.3) = -3.2c + 13.76 So, our inequality becomes 20≥−3.2c+13.7620 \geq -3.2c + 13.76.

STEP 4

Now, we want to get the term with cc by itself.
We can do this by **subtracting** 13.7613.76 from both sides of the inequality.
Remember, what we do to one side, we *must* do to the other! 20−13.76≥−3.2c+13.76−13.7620 - 13.76 \geq -3.2c + 13.76 - 13.76 6.24≥−3.2c6.24 \geq -3.2c

STEP 5

To **isolate** cc, we need to **divide** both sides of the inequality by −3.2-3.2. *But remember*, when we divide by a negative number, we have to **flip** the inequality sign! 6.24−3.2≤−3.2c−3.2\frac{6.24}{-3.2} \leq \frac{-3.2c}{-3.2} −1.95≤c-1.95 \leq c

STEP 6

We can rewrite −1.95≤c-1.95 \leq c as c≥−1.95c \geq -1.95.
This just makes it a little easier to read and understand.
It means the same thing!

STEP 7

Draw a number line and mark −1.95-1.95 on it.

STEP 8

Since our inequality is c≥−1.95c \geq -1.95, we want to include −1.95-1.95 in our solution.
We show this with a **closed circle** at −1.95-1.95.
Then, since cc can be any number *greater than or equal to* −1.95-1.95, we draw a **line** to the *right* of −1.95-1.95 with an arrow to show that it continues forever!

STEP 9

The solution is c≥−1.95c \geq -1.95.
The graph is a number line with a closed circle at −1.95-1.95 and a line extending to the right.

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