Inequalities and Their Solutions

Professor Dave explains how to graph inequalities, starting with a review of basic inequality concepts. He demonstrates solving polynomial inequalities by rearranging terms and testing intervals on a number line. The video then covers graphing two-variable inequalities by treating them as equations and shading the solution area on a coordinate plane. Finally, rational inequalities are addressed by finding intercepts and asymptotes, sketching the function, and determining intervals where the function is positive.

What is the main difference between an equation and an inequality?

• An equation shows equivalence, while an inequality shows a range of values.

• An equation is always true, while an inequality is always false.

• An equation has variables, while an inequality does not.

• An equation is solved graphically, while an inequality is solved algebraically.

If X is greater than or equal to 3, which of the following is NOT a possible value for X?

• 4

• 2.999

• 3.001

• 3

What is the first step in solving the inequality X^2 + 6X < -5?

• Multiply both sides by 2

• Subtract 6X from both sides

• Add 5 to both sides

• Divide both sides by X

When solving X^2 + 6X < -5, what are the solutions after factoring?

• X = 2 and X = 3

• X = 0 and X = 6

• X = -1 and X = -5

• X = 1 and X = 5

How do you determine which sections of the number line satisfy the inequality?

• By solving for X in each section

• By testing a point from each section

• By graphing the inequality

• By using a calculator

What is the solution to the inequality X^2 + 6X < -5?

• From -5 to -1, inclusive

• From -5 to -1, exclusive

• From -1 to 5, inclusive

• From -1 to 5, exclusive

When graphing Y ≥ X + 2, what does the shaded region represent?

• Points on and below the line

• Points below the line

• Points on the line

• Points above the line

In the inequality Y ≥ X + 2, what does the line Y = X + 2 represent?

• The boundary of the solution region

• The solution region itself

• The maximum value of Y

• The axis of symmetry

How do you graph the inequality Y > X^2 - 4?

• Shade to the left of the parabola

• Shade on the parabola

• Shade above the parabola

• Shade below the parabola

What is the purpose of shading in the graph of an inequality?

• To indicate the solution region

• To highlight the axis

• To show the line of symmetry

• To mark the intercepts