aInequalities can be solved in the same way as equations, by performing inverse operations on both sides until the variable is isolated. The only difference is that when you divide or multiply by a negative number, you must reverse the inequality sign.
This inequality tells us that all values greater than or equal to 4 will satisfy the inequality.
Below we demonstrate the inequality by graphing the solution set on a number line. Notice that x cannot equal 4, which we show with an open circle on the number line.
b Inequalities can be solved in the same way as equations, by performing inverse operations on both sides until the variable is isolated. The only difference is that when you divide or multiply by a negative number, you must reverse the inequality sign.
This inequality tells us that all values greater than 20.5 will satisfy the inequality.
Below we demonstrate the inequality by graphing the solution set on a number line. Notice that x cannot equal 20.5, which we show with an open circle on the number line.
c We will begin by moving all the variable terms to one side of the equation.
x^2 ≤ - 4x + 5 ⇔ 0 ≤ - x^2 - 4x + 5
Next, we will sketch the related quadratic function. To do so, we first need to identify the values of a, b, and c.
y = - x^2 - 4x + 5 ⇔ y= - 1x^2 + ( -4)x + 5We see that a= - 1, b= -4, and c= 5. Let's substitute these values into the Quadratic Formula to find the roots of 0= - x^2-4x+5=.
Now we can calculate the first root using the positive sign and the second root using the negative sign.
x=4 ± 6/-2
x=4 + 6/-2
x=4 - 6/-2
x=10/-2
x=-2/-2
x= -5
x=1
The solution of the given quadratic inequality, 0 ≤ ax^2+bx+c, consists of x-values for which the graph of the related quadratic function lies on and above the x-axis. The graph opens downward, since a= - 1 is less than zero.
We see that the graph lies on and below the x-axis from x = -5 to x = 1.
-5 ≤ x ≤ 1
In order to show the solution set on the number line, we only need to take the x-axis from the previous graph. The blue segment shows the solution set.
