solv quadiratic inequalities using product propertise
2xsq<x+2
To solve quadratic inequalities, using the product properties, you'll need to follow these steps:
Step 1: Rewrite the quadratic inequality in the form ax^2 + bx + c < 0. In your case, you have 2x^2 < x + 2.
Step 2: Set the quadratic expression equal to zero. In this case, it becomes 2x^2 - x - 2 = 0.
Step 3: Solve the quadratic equation by factoring or using the quadratic formula. The factored form of 2x^2 - x - 2 = 0 is (2x + 1)(x - 2) = 0. So we have two possible solutions: 2x + 1 = 0 or x - 2 = 0.
Step 4: Solve for x in each equation. For 2x + 1 = 0, subtract 1 from both sides and divide by 2, giving you x = -1/2. For x - 2 = 0, add 2 to both sides, yielding x = 2.
Step 5: Plot the solutions on the number line. We have two critical points: -1/2 and 2.
Step 6: Create test intervals by selecting points between and outside the critical points.
- Test a point less than -1/2, such as x = -2. Plugging this value into the quadratic inequality, we get 2*(-2)^2 < -2 + 2, which simplifies to 8 < 0, which is false.
- Test a point between -1/2 and 2, such as x = 0. Plugging this value into the quadratic inequality, we get 2*(0)^2 < 0 + 2, which simplifies to 0 < 2, which is true.
- Test a point greater than 2, such as x = 3. Plugging this value into the quadratic inequality, we get 2*(3)^2 < 3 + 2, which simplifies to 18 < 5, which is false.
Step 7: Determine the solution based on the test intervals. Since the inequality states that 2x^2 < x + 2, the solution is x values that satisfy the quadratic inequality, which are x < -1/2 or 0 < x < 2.
Therefore, the solution to the quadratic inequality 2x^2 < x + 2, using the product properties, is x < -1/2 or 0 < x < 2.