1) To solve the inequality x² + 3x - 4 ≥ 0, we first need to factor the quadratic equation x² + 3x - 4.
x² + 3x - 4 can be factored as (x + 4)(x - 1).
Now, we can use the zero product property to find the critical points by setting each factor equal to zerox + 4 = 0 or x - 1 = x = -4 x = 1
Now we can create a number line and test the intervals between the critical points (-4 and 1) to see where the inequality holds true.
For x < -4, the inequality does not hold trueFor -4 < x < 1, the inequality does hold trueFor x > 1, the inequality also holds true.
Therefore, the solution to the inequality x² + 3x - 4 ≥ 0 is x ≤ -4 or x ≥ 1.
2) To solve the inequality -x² + 10x - 16 > 0, we can first rewrite it as x² - 10x + 16 < 0.
(x - 8)(x - 2) < 0
Now set each factor equal to zero to find the critical pointsx - 8 = 0 or x - 2 = x = 8 x = 2
On the number line, we can test the intervals between the critical points (2 and 8) to see where the inequality holds true.
For x < 2 or x > 8, the inequality does not hold trueFor 2 < x < 8, the inequality does hold true.
Therefore, the solution to the inequality -x² + 10x - 16 > 0 is 2 < x < 8.
3) To solve the inequality x² ≤ 81, we need to take the square root of both sides to remove the square.
√(x²) ≤ √8|x| ≤ 9
This simplifies to -9 ≤ x ≤ 9.
Therefore, the solution to the inequality x² ≤ 81 is -9 ≤ x ≤ 9.
1) To solve the inequality x² + 3x - 4 ≥ 0, we first need to factor the quadratic equation x² + 3x - 4.
x² + 3x - 4 can be factored as (x + 4)(x - 1).
Now, we can use the zero product property to find the critical points by setting each factor equal to zero
x + 4 = 0 or x - 1 =
x = -4 x = 1
Now we can create a number line and test the intervals between the critical points (-4 and 1) to see where the inequality holds true.
For x < -4, the inequality does not hold true
For -4 < x < 1, the inequality does hold true
For x > 1, the inequality also holds true.
Therefore, the solution to the inequality x² + 3x - 4 ≥ 0 is x ≤ -4 or x ≥ 1.
2) To solve the inequality -x² + 10x - 16 > 0, we can first rewrite it as x² - 10x + 16 < 0.
(x - 8)(x - 2) < 0
Now set each factor equal to zero to find the critical points
x - 8 = 0 or x - 2 =
x = 8 x = 2
On the number line, we can test the intervals between the critical points (2 and 8) to see where the inequality holds true.
For x < 2 or x > 8, the inequality does not hold true
For 2 < x < 8, the inequality does hold true.
Therefore, the solution to the inequality -x² + 10x - 16 > 0 is 2 < x < 8.
3) To solve the inequality x² ≤ 81, we need to take the square root of both sides to remove the square.
√(x²) ≤ √8
|x| ≤ 9
This simplifies to -9 ≤ x ≤ 9.
Therefore, the solution to the inequality x² ≤ 81 is -9 ≤ x ≤ 9.