Solve the following inequality. Write the solution in interval notation_ x2 3x – 4 > 0
The Correct Answer and Explanation is:
We are asked to solve the inequality:x2+3x−4>0x^2 + 3x – 4 > 0x2+3x−4>0
Step 1: Solve the Related Equation
To solve the inequality, first find the roots of the corresponding equation:x2+3x−4=0x^2 + 3x – 4 = 0x2+3x−4=0
Use the quadratic formula:x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}x=2a−b±b2−4ac
Here, a=1a = 1a=1, b=3b = 3b=3, and c=−4c = -4c=−4. Plugging into the formula:x=−3±32−4(1)(−4)2(1)=−3±9+162=−3±252x = \frac{-3 \pm \sqrt{3^2 – 4(1)(-4)}}{2(1)} = \frac{-3 \pm \sqrt{9 + 16}}{2} = \frac{-3 \pm \sqrt{25}}{2}x=2(1)−3±32−4(1)(−4)=2−3±9+16=2−3±25x=−3±52x = \frac{-3 \pm 5}{2}x=2−3±5
This gives two real roots:x=−3+52=1andx=−3−52=−4x = \frac{-3 + 5}{2} = 1 \quad \text{and} \quad x = \frac{-3 – 5}{2} = -4x=2−3+5=1andx=2−3−5=−4
Step 2: Determine the Sign of the Expression in Each Interval
The roots divide the number line into three intervals:
- (−∞,−4)(-\infty, -4)(−∞,−4)
- (−4,1)(-4, 1)(−4,1)
- (1,∞)(1, \infty)(1,∞)
Choose a test point in each interval to determine the sign of the expression x2+3x−4x^2 + 3x – 4×2+3x−4.
- Interval 1: x=−5x = -5x=−5 (−5)2+3(−5)−4=25−15−4=6>0(-5)^2 + 3(-5) – 4 = 25 – 15 – 4 = 6 > 0(−5)2+3(−5)−4=25−15−4=6>0
- Interval 2: x=0x = 0x=0 02+3(0)−4=−4<00^2 + 3(0) – 4 = -4 < 002+3(0)−4=−4<0
- Interval 3: x=2x = 2x=2 22+3(2)−4=4+6−4=6>02^2 + 3(2) – 4 = 4 + 6 – 4 = 6 > 022+3(2)−4=4+6−4=6>0
Step 3: Interpret the Results
We want the expression to be greater than zero, so we select the intervals where the expression is positive. From our testing:
- Positive in (−∞,−4)(-\infty, -4)(−∞,−4) and (1,∞)(1, \infty)(1,∞)
- Not included at x=−4x = -4x=−4 and x=1x = 1x=1 because the inequality is strictly greater than zero and not inclusive
Final Answer in Interval Notation
(−∞,−4)∪(1,∞)\boxed{(-\infty, -4) \cup (1, \infty)}(−∞,−4)∪(1,∞)
Explanation
To solve the inequality x2+3x−4>0x^2 + 3x – 4 > 0x2+3x−4>0, we begin by analyzing the related quadratic equation. The key is to understand where the quadratic expression is positive. First, we find the roots of the equation x2+3x−4=0x^2 + 3x – 4 = 0x2+3x−4=0 using the quadratic formula. After simplifying, we find the roots are x=−4x = -4x=−4 and x=1x = 1x=1. These roots divide the number line into three intervals: less than -4, between -4 and 1, and greater than 1.
Next, we pick a test value in each interval to check whether the expression is positive or negative in that section. For example, choosing x=−5x = -5x=−5 in the interval less than -4, we substitute into the original expression and find that it results in a positive value. This indicates the expression is greater than zero on that interval. Doing the same for the other two intervals, we find that the expression is negative between -4 and 1, and again positive when x>1x > 1x>1.
Since the original inequality asks for where the expression is strictly greater than zero, we exclude the values x=−4x = -4x=−4 and x=1x = 1x=1, because the expression equals zero at those points, not greater than zero. Therefore, the final solution includes only the intervals where the expression is strictly positive: from negative infinity to -4, and from 1 to positive infinity.
This approach works for any quadratic inequality: find the roots, use test intervals to check signs, and determine which regions satisfy the inequality. Express the final answer in interval notation, excluding boundary points for strict inequalities.
