Explain Simultaneous inequality With example x^2-4x+2>-2x^2+7x-3>-x^2+4
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A simultaneous inequality is a compound inequality where multiple conditions must be satisfied at the same time. The general form is a less than f of x less than b, which means f of x is greater than a AND f of x is less than b. For example, if we have 2 less than x plus 1 less than 5, we need to find values of x where x plus 1 is both greater than 2 and less than 5. This creates a solution interval from 1 to 4, shown as the green region on the graph.
Let's break down the given simultaneous inequality x squared minus 4x plus 2 is greater than negative 2x squared plus 7x minus 3, which is greater than negative x squared plus 4. This represents two separate inequalities that must both be true. First, x squared minus 4x plus 2 must be greater than negative 2x squared plus 7x minus 3. Second, negative 2x squared plus 7x minus 3 must be greater than negative x squared plus 4. We rearrange these to standard form: the first becomes 3x squared minus 11x plus 5 greater than 0, and the second becomes negative x squared plus 7x minus 7 greater than 0.
Now let's solve the first inequality: 3x squared minus 11x plus 5 greater than 0. Using the quadratic formula, we find the roots are x equals 11 minus square root of 61 over 6, approximately 0.47, and x equals 11 plus square root of 61 over 6, approximately 3.53. Since the coefficient of x squared is positive, the parabola opens upward. This means the quadratic expression is positive when x is less than 0.47 or when x is greater than 3.53. The green shaded regions on the graph show where the inequality is satisfied.
Now let's solve the second inequality: negative x squared plus 7x minus 7 greater than 0, which is equivalent to x squared minus 7x plus 7 less than 0. Using the quadratic formula, we find the roots are x equals 7 minus square root of 21 over 2, approximately 1.21, and x equals 7 plus square root of 21 over 2, approximately 5.79. Since we need the quadratic to be negative, and the parabola opens upward, the solution is between the roots: 1.21 less than x less than 5.79. The green shaded region shows where this inequality is satisfied.
Now we find the intersection of both solutions. The first inequality gives us x less than 0.47 or x greater than 3.53. The second inequality gives us 1.21 less than x less than 5.79. The intersection of x less than 0.47 with the interval from 1.21 to 5.79 is empty, since these ranges don't overlap. However, the intersection of x greater than 3.53 with the interval from 1.21 to 5.79 gives us 3.53 less than x less than 5.79. We can verify this by testing x equals 4: we get 2 greater than negative 3 greater than negative 12, which is true.