solve the inequality algebraically

11x-3 is greater than or equal to - 4x^2

11x - 3 ≥ -4x^2

4x^2 + 11x - 3 > 0
(x+3)(4x-1) ≥ 0

critical values are x = -3 and x = 1/4
and y = 4x^2 + 11x - 3 is a parabola opening upwards, with intercepts -3 and 1/4 .....

so x ≤ - 3 OR x ≥ 1/4

To solve the inequality algebraically, follow these steps:

Step 1: Rearrange the inequality by moving all terms to one side to set it equal to zero:
-4x^2 + 11x - 3 ≥ 0

Step 2: Put the quadratic expression in standard form (highest power to lowest power):
-4x^2 + 11x - 3 ≥ 0

Step 3: To solve the inequality, we can find the x-intercepts or the critical points of the quadratic equation. So, let's factor it:
-4x^2 + 11x - 3 ≥ 0
(-4x + 3)(x - 1) ≥ 0

Step 4: Set each factor greater than or equal to zero and solve for x:
-4x + 3 ≥ 0
x - 1 ≥ 0

Solving these individually:
-4x + 3 ≥ 0
-4x ≥ -3
x ≤ -3/-4
x ≤ 3/4

x - 1 ≥ 0
x ≥ 1

Step 5: Now, we determine the sign of the inequality expression between the critical points (3/4 and 1). We can use a number line and test a value from each interval.

Choose a test point from each interval:
For x < 3/4: Test x = 0
-4(0) + 3 > 0 (negative result)
-3 > 0 (False)

For x between 3/4 and 1: Test x = 1
-4(1) + 3 > 0 (positive result)
-4 + 3 > 0
-1 > 0 (False)

For x > 1: Test x = 2
-4(2) + 3 > 0 (positive result)
-8 + 3 > 0
-5 > 0 (False)

Step 6: By analyzing the signs, we can determine the solution to the inequality.

From the test points, we can see that the inequality is true when:
-4x + 3 ≤ 0, which means x ≤ 3/4
x - 1 ≥ 0, which means x ≥ 1

Combining the two conditions, the solution is:
x ≤ 3/4 or x ≥ 1

So, the inequality solution is x ≤ 3/4 or x ≥ 1.