x^2-2x-3 ≤ 0
⇕
x^2-2x ≤ 3
In a quadratic expression, b is the linear coefficient. For the inequality above, we have that b=- 2. Let's now calculate ( b2 )^2.
Now we can take the square root of the inequality. Remember that we do not know the sign of x. Thus, to consider both possibilities we need to use the absolute value.
Now we can create a compound inequality by removing the absolute value. In this case, the solution set is any number less than or equal to 2 away from the midpoint in the positive direction and any number less than or equal to 2 away from the midpoint in the negative direction.
Absolute Value Inequality:& |x-1| ≤ 2
Compound Inequality:& - 2≤ x-1 ≤ 2
We can split this compound inequality into two cases: one where x-1 is greater than or equal to -2 and one where x-1 is less than or equal to 2.
x-1 ≥ - 2 and x-1 ≤ 2
Let's isolate x in both of these cases.
This inequality tells us that all values greater than or equal to - 1 will satisfy the inequality.
Solution Set
The solution to this type of compound inequality is the overlap of the solution sets. Let's recombine our cases back into one compound inequality.
First Solution Set:& x ≤ 3
Second Solution Set:& - 1 ≤ x
Intersecting Solution Set:& - 1 ≤ x ≤ 3
b We want to solve the quadratic equation by completing the square. Note that all terms with x are on one side of the equation.
x^2+4x=3In a quadratic expression b is the linear coefficient. For the equation above, we have that b=4. Let's now calculate ( b2 )^2.
Both x=- 2+ sqrt(7) and x=- 2- sqrt(7) are solutions of the equation.
c We want to solve the quadratic inequality by completing the square. Let's start by rewriting the inequality so all terms with x are on one side of the inequality and all constants are on the other side.
x^2+5x-36 > 0
⇕
x^2+5x > 36
In a quadratic expression, b is the linear coefficient. For the inequality above, we have that b=5. Let's now calculate ( b2 )^2.
Now we can take the square root of the inequality. Remember that we do not know the sign of x. Thus, to consider both possibilities we need to use the absolute value.
Now we will create a compound inequality by removing the absolute value. In this case, and since x+2.5 can be written as x-(- 2.5), the solution set contains the numbers that make the distance between x and - 2.5 greater than 6.5 in the positive direction or in the negative direction.
x+2.5 > 6.5 or x+2.5 < -6.5
Let's isolate x in both of these cases.
This inequality tells us that all values less than - 9 will satisfy the inequality.
Solution Set
The solution to this type of compound inequality is the combination of the solution sets.
First Solution Set:& x > 4
Second Solution Set:& x < -9
Combined Solution Set:& x < -9 or x > 4
d We want to solve the quadratic equation by completing the square. To do so, we will start by rewriting the equation so all terms with x are on one side of the equation and all constants are on the other side.
x^2-3x-13.75=0
⇕
x^2-3x=13.75In a quadratic expression b is the linear coefficient. For the equation above, we have that b=- 3. Let's now calculate ( b2 )^2.
