Find the no. Of integers satisfying the inequality(I) x²+6x-7 is less than or equal to 2

To solve the inequality \( x^2 + 6x - 7 \leq 2 \), we first need to rearrange the expression to set it to zero. This allows us to find the critical points, which are essential in determining the intervals where the inequality holds true.

Rearranging the Inequality

Start by subtracting 2 from both sides:

\( x^2 + 6x - 7 - 2 \leq 0 \)

Which simplifies to:

\( x^2 + 6x - 9 \leq 0 \)

Finding the Roots

Next, we need to find the roots of the equation \( x^2 + 6x - 9 = 0 \). We can use the quadratic formula, where \( a = 1 \), \( b = 6 \), and \( c = -9 \):

\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Calculating the discriminant:

\( b^2 - 4ac = 6^2 - 4 \cdot 1 \cdot (-9) = 36 + 36 = 72 \)

Now substituting into the formula:

\( x = \frac{-6 \pm \sqrt{72}}{2} \)

We can simplify \( \sqrt{72} \) to \( 6\sqrt{2} \), so:

\( x = \frac{-6 \pm 6\sqrt{2}}{2} = -3 \pm 3\sqrt{2} \)

Identifying Critical Points

The critical points are:

\( x_1 = -3 - 3\sqrt{2} \) and \( x_2 = -3 + 3\sqrt{2} \)

These points will help us determine the intervals on the number line where the inequality holds true.

Testing Intervals

Now we need to test the intervals formed by these critical points:

• Interval 1: \( (-\infty, -3 - 3\sqrt{2}) \)
• Interval 2: \( (-3 - 3\sqrt{2}, -3 + 3\sqrt{2}) \)
• Interval 3: \( (-3 + 3\sqrt{2}, \infty) \)

We can pick test points from each interval to see if the inequality holds. For example:

• Choose \( x = -10 \) for Interval 1:

\( (-10)^2 + 6(-10) - 9 = 100 - 60 - 9 = 31 \) (not less than or equal to 0)

• Choose \( x = -3 \) for Interval 2:

\( (-3)^2 + 6(-3) - 9 = 9 - 18 - 9 = -18 \) (less than or equal to 0)

• Choose \( x = 0 \) for Interval 3:

\( 0^2 + 6(0) - 9 = -9 \) (not less than or equal to 0)

Conclusion on Valid Intervals

From this testing, we can conclude that the inequality \( x^2 + 6x - 9 \leq 0 \) is satisfied in the interval:

\( [-3 - 3\sqrt{2}, -3 + 3\sqrt{2}] \)

Calculating the Integer Solutions

Now we need to find the integers within this interval. We can approximate the critical points:

\( 3\sqrt{2} \) is approximately \( 4.24 \), so:

• \( -3 - 3\sqrt{2} \approx -7.24 \)
• \( -3 + 3\sqrt{2} \approx 1.24 \)

This means the integers in the interval are:

• -7
• -6
• -5
• -4
• -3
• -2
• -1
• 0
• 1

Counting these, we find there are a total of 9 integers satisfying the original inequality.