L8 Quadratic Inequalities - Part 4
Completion requirements
Unit 1A
Precalculus
Lesson 8: Quadratic Inequalities
- Sign Analysis Method
One last time, consider the inequality determined from the scenario described at the beginning of this Lesson. Factor the quadratic and simplify both sides of the inequality by multiplying by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle
mathsize=¨14px¨»«mrow»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mstyle»«/math».
The roots of the corresponding quadratic equation «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»0«/mn»«mo»=«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«mi»)(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»20«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»20«/mn»«/mrow»«/mstyle»«/math». These values act as the boundaries for sign analysis.
When values of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#60;«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math» are substituted into the factor «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», the value of the factor will be negative. When values of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#62;«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math» are substituted into the factor «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», the value of the factor will be positive. These results are summarized on the line graph below.
Similarly, the values that make the factor «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»20«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» positive and negative must also be considered.
When values of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#60;«/mo»«mn»20«/mn»«/mrow»«/mstyle»«/math» are substituted into the factor «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»20«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», the value of the factor will be negative.
When values of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#62;«/mo»«mn»20«/mn»«/mrow»«/mstyle»«/math» are substituted into the factor «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»20«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», the value of the factor will be positive.
When the two factors are multiplied together, the product will have a positive sign when both factors are negative or when both factors are positive. The product will have a negative sign when the signs of the two factors are different, that is, one is positive and one is negative. The number line below shows the signs of the product «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«mi»)(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»20«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math».
Use the last number line to state the solution to the inequality «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«mi»)(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»20«/mn»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8804;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math». The solution to this inequality is shown where the product of the two factors is less than or equal to zero. That is, where the product is negative. The number line shows the product is negative between «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»20«/mn»«/mstyle»«/math». The inequality symbol indicates «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»20«/mn»«/mstyle»«/math» are also included in the solution. Therefore, the solution is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mn»5«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»20«/mn»«/mrow»«/mfenced»«/mstyle»«/math».
Alternative Solution
Sign analysis can also be summarized in a chart. The roots of the corresponding quadratic equation are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»=«/mo»«mn»20«/mn»«/mstyle»«/math». These values form the boundaries for three intervals requiring attention, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»§#8722;«/mo»«mo»§#8734;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»5«/mn»«/mrow»«/mfenced»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»5«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»20«/mn»«/mrow»«/mfenced»«/mstyle»«/math», and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»20«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8734;«/mo»«/mrow»«/mfenced»«/mstyle»«/math». For each of these intervals, determine the signs of the factors, and then work down the column to find the product.
Use the bottom row to determine the solution to the inequality «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«mi»)(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»20«/mn»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8804;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math». The solution to this inequality is shown where the product of the two factors is less than or equal to zero. That is, where the product is zero or negative. From the chart, the only interval in which the product is negative is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»5«/mn»«mo»§#60;«/mo»«mi»x«/mi»«mo»§#60;«/mo»«mn»20«/mn»«/mrow»«/mstyle»«/math». The inequality symbol indicates «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»20«/mn»«/mstyle»«/math» are also included in the solution. Therefore, the solution is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»5«/mn»«mo»§#8804;«/mo»«mi»x«/mi»«mo»§#8804;«/mo»«mn»20«/mn»«/mrow»«/mstyle»«/math» or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mn»5«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»20«/mn»«/mrow»«/mfenced»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mo»§#8722;«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«mi»)(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»20«/mn»«mi
mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»§#8805;«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«mi»)(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»20«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mtd»«mtd»«mo»§#8804;«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The roots of the corresponding quadratic equation «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»0«/mn»«mo»=«/mo»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«mi»)(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»20«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»20«/mn»«/mrow»«/mstyle»«/math». These values act as the boundaries for sign analysis.
When values of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#60;«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math» are substituted into the factor «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», the value of the factor will be negative. When values of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#62;«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math» are substituted into the factor «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», the value of the factor will be positive. These results are summarized on the line graph below.
Similarly, the values that make the factor «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»20«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math» positive and negative must also be considered.
When values of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#60;«/mo»«mn»20«/mn»«/mrow»«/mstyle»«/math» are substituted into the factor «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»20«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», the value of the factor will be negative.
When values of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#62;«/mo»«mn»20«/mn»«/mrow»«/mstyle»«/math» are substituted into the factor «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»20«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math», the value of the factor will be positive.
When the two factors are multiplied together, the product will have a positive sign when both factors are negative or when both factors are positive. The product will have a negative sign when the signs of the two factors are different, that is, one is positive and one is negative. The number line below shows the signs of the product «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«mi»)(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»20«/mn»«mi mathvariant=¨normal¨»)«/mi»«/mrow»«/mstyle»«/math».
Use the last number line to state the solution to the inequality «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«mi»)(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»20«/mn»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8804;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math». The solution to this inequality is shown where the product of the two factors is less than or equal to zero. That is, where the product is negative. The number line shows the product is negative between «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»20«/mn»«/mstyle»«/math». The inequality symbol indicates «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»20«/mn»«/mstyle»«/math» are also included in the solution. Therefore, the solution is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mn»5«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»20«/mn»«/mrow»«/mfenced»«/mstyle»«/math».
Alternative Solution
Sign analysis can also be summarized in a chart. The roots of the corresponding quadratic equation are «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»=«/mo»«mn»20«/mn»«/mstyle»«/math». These values form the boundaries for three intervals requiring attention, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»§#8722;«/mo»«mo»§#8734;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»5«/mn»«/mrow»«/mfenced»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»5«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»20«/mn»«/mrow»«/mfenced»«/mstyle»«/math», and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»20«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8734;«/mo»«/mrow»«/mfenced»«/mstyle»«/math». For each of these intervals, determine the signs of the factors, and then work down the column to find the product.
| Interval
|
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»§#8722;«/mo»«mo»§#8734;«/mo»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»5«/mn»«/mrow»«/mfenced»«/mstyle»«/math»
or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#60;«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math»
|
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»5«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»20«/mn»«/mrow»«/mfenced»«/mstyle»«/math» or «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»5«/mn»«mo»§#60;«/mo»«mi»x«/mi»«mo»§#60;«/mo»«mn»20«/mn»«/mrow»«/mstyle»«/math»
|
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»20«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mo»§#8734;«/mo»«/mrow»«/mfenced»«/mstyle»«/math» or «math style=¨font-family:Verdana¨
xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#62;«/mo»«mn»20«/mn»«/mrow»«/mstyle»«/math»
|
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»-«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math»
|
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«/mstyle»«/math»
|
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»+«/mo»«/mstyle»«/math»
|
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»+«/mo»«/mstyle»«/math» |
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«mo»-«/mo»«mn»20«/mn»«/mstyle»«/math»
|
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«/mstyle»«/math»
|
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«/mstyle»«/math»
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«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»+«/mo»«/mstyle»«/math» |
| «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mn»5«/mn»«/mrow»«/mfenced»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mn»20«/mn»«/mrow»«/mfenced»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»+«/mo»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»-«/mo»«/mstyle»«/math» | «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»+«/mo»«/mstyle»«/math» |
Use the bottom row to determine the solution to the inequality «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»5«/mn»«mi»)(«/mi»«mi»x«/mi»«mo»§#8722;«/mo»«mn»20«/mn»«mi mathvariant=¨normal¨»)«/mi»«mo»§#8804;«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math». The solution to this inequality is shown where the product of the two factors is less than or equal to zero. That is, where the product is zero or negative. From the chart, the only interval in which the product is negative is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»5«/mn»«mo»§#60;«/mo»«mi»x«/mi»«mo»§#60;«/mo»«mn»20«/mn»«/mrow»«/mstyle»«/math». The inequality symbol indicates «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»20«/mn»«/mstyle»«/math» are also included in the solution. Therefore, the solution is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»5«/mn»«mo»§#8804;«/mo»«mi»x«/mi»«mo»§#8804;«/mo»«mn»20«/mn»«/mrow»«/mstyle»«/math» or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced open=¨[¨ close=¨]¨»«mrow»«mn»5«/mn»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mn»20«/mn»«/mrow»«/mfenced»«/mstyle»«/math».
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