The values of the equation are:1) the solution to the inequality 3x² - 5x - 2 ≤ 0 is x ≤ -1/3 or x = 0. (2)the solution to the inequality 6y² + 5y > 4 is y > 1/2 and y > -4/3. (3) the solution to the inequality 4x² - 4x - 15 ≥ 0 is x ≥ 5/2 and x ≥ -3/2. (4) the solution to the inequality 4w² - 4w - 3 > 0 is w < -1/2 and w > 3/2.
What is an equation?
To factor 3x² - 5x - 2,
we get: (3x + 1)(x - 2).
Set each factor equal to zero: To find the critical points, we set each factor equal to zero:
- 3x + 1 = 0 --> x = -1/3
- x - 2 = 0 --> x = 2
Plot the critical points on a number line:
-∞ |-----|-----|-----| +∞
-1/3 2
- For x = -1: Substitute -1 into the inequality: 3(-1)² - 5(-1) - 2 ≤ 0
Simplifying, we get: 3 + 5 - 2 ≤ 0 --> 6 ≤ 0
Since 6 is not less than or equal to 0, -1 is not a solution.
- For x = 0: Substitute 0 into the inequality: 3(0)² - 5(0) - 2 ≤ 0
Simplifying, we get: -2 ≤ 0
Since -2 is less than or equal to 0, 0 is a solution.
Therefore, the solution to the inequality 3x² - 5x - 2 ≤ 0 is x ≤ -1/3 or x = 0.
2) Rewrite the inequality in standard quadratic form: Rearrange the inequality to have the quadratic expression on one side and 0 on the other side. We get: 6y² + 5y - 4 > 0.
2. Factor the quadratic expression: To factor 6y² + 5y - 4, we need to find two binomials that multiply together to give this expression. After factoring, we get: (2y - 1)(3y + 4).
3. Set each factor greater than zero: To find the critical points, we set each factor greater than zero:
- 2y - 1 > 0 --> y > 1/2
- 3y + 4 > 0 --> y > -4/3
4. Plot the critical points on a number line: We place the critical points (1/2 and -4/3) on a number line, dividing it into three intervals.
-∞ |-----|-----|-----| +∞
-4/3 1/2
5. Determine the solution intervals: Since the inequality is greater than 0, we need to find the intervals where the expression is positive.
- For y > 1/2: All values of y greater than 1/2 satisfy the inequality.
- For y > -4/3: All values of y greater than -4/3 also satisfy the inequality.
6. Determine the final solution: Combining the solution intervals, we find that the solutions to the inequality are y > 1/2 and y > -4/3.
Therefore, the solution to the inequality 6y² + 5y > 4 is y > 1/2 and y > -4/3.
(3) Factor the quadratic expression: To factor 4x² - 4x - 15, we need to find two binomials that multiply together to give this expression. After factoring, we get: (2x - 5)(2x + 3).
2. Set each factor greater than or equal to zero: To find the critical points, we set each factor greater than or equal to zero:
- 2x - 5 ≥ 0 --> x ≥ 5/2
- 2x + 3 ≥ 0 --> x ≥ -3/2
3. Plot the critical points on a number line: We place the critical points (5/2 and -3/2) on a number line, dividing it into three intervals.
-∞ |-----|-----|-----| +∞
-3/2 5/2
4. Determine the solution intervals: Since the inequality is greater than or equal to 0, we need to find the intervals where the expression is non-negative.
- For x ≥ 5/2: All values of x greater than or equal to 5/2 satisfy the inequality.
- For x ≥ -3/2: All values of x greater than or equal to -3/2 also satisfy the inequality.
5. Determine the final solution: Combining the solution intervals, we find that the solutions to the inequality are x ≥ 5/2 and x ≥ -3/2.
Therefore, the solution to the inequality 4x² - 4x - 15 ≥ 0 is x ≥ 5/2 and x ≥ -3/2.
(4)
1. Factor the quadratic expression: To factor 4w² - 4w - 3, we need to find two binomials that multiply together to give this expression. However, this quadratic expression cannot be factored easily using integers. So, we'll use the quadratic formula to find the roots of the equation.
2. Apply the quadratic formula: The quadratic formula states that for an equation of the form ax² + bx + c = 0, the solutions for x can be found using the formula: x = (-b ± √(b² - 4ac)) / (2a).
In this case, a = 4, b = -4, and c = -3. Plugging these values into the formula, we get:
w = (-(-4) ± √((-4)² - 4(4)(-3))) / (2(4))
Simplifying, we have:
w = (4 ± √(16 + 48)) / 8
w = (4 ± √64) / 8
w = (4 ± 8) / 8
3. Solve for the two possible values of w: Using the values obtained from the quadratic formula, we have two solutions:
- w = (4 + 8) / 8 = 12 / 8 = 3/2
- w = (4 - 8) / 8 = -4 / 8 = -1/2
4. Determine the solution intervals: Since the inequality is greater than zero, we need to find the intervals where the expression is positive.
- For w < -1/2: All values of w less than -1/2 satisfy the inequality.
- For -1/2 < w < 3/2: All values of w between -1/2 and 3/2 do not satisfy the inequality.
- For w > 3/2: All values of w greater than 3/2 satisfy the inequality.
5. Determine the final solution: Combining the solution intervals, we find that the solutions to the inequality 4w² - 4w - 3 > 0 are w < -1/2 and w > 3/2.
Therefore, the solution to the inequality 4w² - 4w - 3 > 0 is w < -1/2 and w > 3/2.