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Wize High School Grade 12 Pre-Calculus Textbook > Inequalities

Solving Polynomial Inequalities (Graphs)

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Solving Polynomial Inequalities Using a Graph

A polynomial inequality can be solved in similar ways the polynomial equations are solved.

Step 1.
Factor the polynomial.

Step 2.
Identify the x-intercepts and graph the polynomial.

Step 3.
Identify the intervals where the polynomial exists to satisfy the inequality.

Step 4.
Express the answer in an appropriate format.

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Example

Solve x3−2x2−5x+6≤0x^3-2x^2-5x+6\leq{}0 by graphing. Express the solution in set notation.

Step 1.
Let x=1x=1.

1−2−561↓1−1−61−1−60\begin{array}{r|cccc} &1&-2&-5&6\\ 1&\downarrow&1&-1&-6\\\hline &1&-1&-6&0 \end{array}

The quotient becomes x2−x−6=(x−3)(x+2)x^2-x-6=(x-3)(x+2).

So, x3−2x2−5x+6=(x−1)(x+2)(x−3)≤0x^3-2x^2-5x+6=(x-1)(x+2)(x-3)\leq{}0

Step 2.
X-intercepts: 1,−2,31, -2, 3
Graph:

Step 3.

The polynomial is less than or equals to 0. Thus, we want the part of the polynomial that lies below the line y = 0 (the x-axis).

The intervals where the polynomial is less than 0 are (−∞,2] ∪ [1,3](-\infin,2]~\cup~[1,3]

Step 4.

(−∞,2] ∪ [1,3](-\infin,2]~\cup~[1,3] in set notation is:

{x∈R∣ x≤−2  &  1≤x≤3}\{x\in\mathbb{R}|~x\leq{}-2~~\&~~1\leq{}x\leq{}3\}.
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Example: Solving Polynomial Inequalities Using a Graph

Solve x3−x2+x−1≥0x^3-x^2+x-1\geq{}0 by graphing. Express the solution in interval notation.

Step 1.
Let x=1x=1.

1−11−11↓1011010\begin{array}{r|cccc} &1&-1&1&-1\\ 1&\downarrow&1&0&1\\\hline &1&0&1&0 \end{array}

The quotient becomes x2+1x^2+1.

So, x3−x2+x−1=(x−1)(x2+1)≥0x^3-x^2+x-1=(x-1)(x^2+1)\geq{}0

Step 2.
X-intercepts: −1-1
Graph:

Step 3.

The polynomial is greater than or equals to 0. Thus, we want the part of the polynomial that lies above the line y = 0 (the x-axis).

The interval where the polynomial is greater than 0 is [1,∞)[1,\infin)

Step 4.
[1,∞)[1,\infin)

Practice: Solving Polynomial Inequalities Using a Graph


Match the graphs to an inequality.
A.
B.
C.
D.
y≥0.01(2x+1)(4x−5)(4x−3)y\geq{}0.01(2x+1)(4x-5)(4x-3)
y<(2x−3)(x+4)y<(2x-3)(x+4)
y>(3x+1)(5−x)y>(3x+1)(5-x)
y≤0.05(6x−5)(4−9x)(x+1)y\leq{}0.05(6x-5)(4-9x)(x+1)

Practice: Solving Polynomial Inequalities Using a Graph


Solve x3−4x2+x+6≤0x^3-4x^2+x+6\leq{}0 by graphing.

Practice: Solving Polynomial Inequalities Using a Graph


Determine an expression for a polynomial, f(x),f(x), that has the following properties:
  • f(x)≤0f(x)\leq{}0 when x≤−5x\leq{}-5
  • f(x)≥0f(x)\geq{}0 when x≥−5x\geq{}-5
  • There is an x-intercept at x=3x=3 with mulitiplicity of 2.
  • f(0)=5f(0)=5
Express f(x) f(x)~ in factored form a(x−x1)(x−x2)...(x−xn)a(x-x_1)(x-x_2)...(x-x_n)
Extra Practice
Solving Polynomial Inequalities Using a Graph
Practice: Solving Polynomial Inequalities Using a Graph
Determine an expression for a polynomial, f(x),f(x), that has the following properties:
f(x)<0f(x)<0 when x<−3  &  0<x<4x<-3~~\&~~0<x<4
Solving Polynomial Inequalities Using a Graph
Practice: Solving Polynomial Inequalities Using a Graph
Determine an expression for a polynomial, f(x),f(x), that has the following properties:
f(x)≤0f(x)\leq{}0 when x≥−7x\geq{}-7
Solving Polynomial Inequalities Using a Graph
Practice: Solving Polynomial Inequalities Using a Graph
Solve x3+7x2−4x−28<0x^3+7x^2-4x-28<0 by graphing.
Solving Polynomial Inequalities Using a Graph
Practice: Solving Polynomial Inequalities Using a Graph
Solve x3+3x2−13x−15≥0x^3+3x^2-13x-15\geq{}0 by graphing.
Solving Polynomial Inequalities Using a Graph
Practice: Solving Polynomial Inequalities Using a Graph
Match the graphs to an inequality.
Solving Polynomial Inequalities Using a Graph
Practice: Solving Polynomial Inequalities Using a Graph
Match the graphs to an inequality.