Our first step will be to solve the given equation for :

Since we want to know the smallest integer of for which , we can set up our equation as

We begin by moving all of our terms to the left side of the inequality.

We then factor.

That means our left side will equal 0 when . However, we also want to know the values when the left side is less than zero. We can do this using test regions. We begin by drawing a number line with our two numbers labeled.

We notice that our two numbers divide our line into three regions. We simply need to try a test value in each region. We begin with our leftmost region by selecting a number less than . We then plug that value into the left side of our inequality to see if the result is positive or negative. Any value (such as ) will give us a positive value.

We then repeat this process with the center region by selecting a value between our two numbers. Any value (such as ) will result in a negative outcome.

Finally we complete the process with the rightmost region by selecting a value larger than . Any value (such as ) will result in a positive value.

We then label our regions accordingly.

Since we want the result to be less than zero, we want the values between our two numbers. However, since our left side can be less than or equal to zero, we can also include the two numbers themselves. We can express this as

To isolate the variable , subtract from both sides of the inequality.

Before we get started, read the question carefully. We need to find x squared, not x. Don't call it quits too early!

So, we start here:

Get the x's on one side and the constants on the other.

When we divide by a negative number in an inequality, remember that we need to switch the direction of the sign.

This problem involves solving the inequality.

Add 3x to both sides

Subtract 7 to each side

divide both sides by7

Begin by simplifying the inequality by using the distributive property.

Add to both sides.

Subtract from both sides.

Because there is a negative variable, multiply both sides by and switch the inequality sign to its opposite.

The first thing we must do is solve the given equation for :

Since we are looking for values when , we can set up our equation as follows:

Solve.

So, is the largest integer of x which makes the statement true.

Add 6 to both sides of the inequality.

To isolate the variable, subtract from both sides of the inequality.