The solution set for polynomial inequality 2x2+5x−3≤0 using a graphing utility.

Question

Jenny Bolton · Accepted Answer

Polynomial inequality is an inequality that can be put in any of the following forms:

f (x) < 0, f (x) > 0, f (x) \leq 0, or f (x) \geq 0

where f is any polynomial function.
A graph technique is used to visualize the solutions of polynomial inequalities. For this, use x-intercepts of given polynomial function f as boundary points that divide the real number line into intervals. On each interval, the graph of f is either above the x-axis

[f (x) > 0]

or below the x-axis

[f (x) < 0]

. This fact gives reasons for x-intercept to play fundamental role in solving polynomial inequalities. The x-intercepts can be found by solving the equation

f (x) = 0

.
Plot the graph of

x^{2} + 3 x - 10 = 0

using the following steps of the Ti-83 calculator:
Step 1: Press the

[Y =]

key: the equations for y will appear.
Step 2: Enter the equations in

Y_{1}

. Here,

Y_{1} = 2 X^{2} + 5 X - 3

.
Step 3: Go to the leftmost side of the line, that is left of

Y_{1}

, and press Enter until you get the sign of >.
Step 4: Press the [Trace] or [GRAPH] key to plot the graph.
Not, look for the points on the x-axis and mention the solution set, that is,

[- 3, \frac{1}{2}]

.
Conclusion: The solution set for the given polynomial inequality

2 x^{2} + 5 x - 3 \leq 0

is

[- 3, \frac{1}{2}]

.

Wendy Boykin · Accepted Answer

Step 1  2x2+5x−3≤0  To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation   ax2+bx+c=a(x−x1)(x−x2),  where x1 and x2 are the solution of the quadratic equation  ax2+bx+c=0  2x2+5x−3=0  All equations of the form  ax2+bx+c=0  can be solved using the quadratic formula:  −b±b2−4ac2a  Substitute 2 for a, 5 for b, and -3 for c in the quadratic formula.  x=−5±52−4×2(−3)2×2  Do the calculations.  x=−5±74  Solve the equation x=−5±74 when ± is plus and when ± is minus  x=12  x=−3  Rewrite the inequality by using the obtained solutions.  2(x−12)(x+3)≤0  For the product to be ≤0, one of the values x−12 and x+3 has to be ≥0 and the other has to be ≤0. Consider the case when x−12≥0 and x+3≤0  x−12≥0  x+3≤0  This is false for any x.  Undefined control sequence cancelUndefined control sequence cancel  Step 2  Consider the case when x−12≤0 and x+3≥0  x+3≥0  x−12≤0  The solution satisfying both inequalities is x∈[−3, 12]  x∈[−3, 12]  The final solution is the union of the obtained solutions.  x∈[−3, 12]

The solution set for polynomial inequality 2x^{2}+5x-3 \le 0 using

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