The set of values of x satisfying the inequalities x^2+3 x+10>0 ; x^2-x-6 ≤0 and 2 x^2+7 x-9 ≥0

The set of values of x satisfying the inequalities x^2+3...

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions where a quadratic polynomial is compared to zero using inequality signs. The solution requires finding the critical points (roots of the corresponding quadratic equation), which divide the real number line into intervals. The sign of the quadratic expression in each interval can then be determined, allowing one to identify the intervals where the inequality holds true.

Factoring Quadratic Equations

Factoring a quadratic equation simplifies the problem of finding its roots, which are the values of the variable for which the quadratic expression becomes zero. These roots are essential for setting up the sign analysis, and they help in determining the intervals on the number line over which the inequality is either satisfied or not.

Sign Analysis

Sign analysis involves testing values from each interval determined by the roots of the quadratic to see whether the quadratic expression yields a positive or negative value. This method helps in identifying which intervals satisfy the original inequality. It is a critical tool for solving quadratic inequalities.

Intersection of Solution Sets

When multiple inequalities are involved, the overall solution is found by taking the intersection of the solution sets for each individual inequality. This means finding the common values that satisfy every inequality simultaneously. This concept is crucial when dealing with compound inequalities in order to arrive at the correct final interval.

Transcript

00:01 So in the given question we are told to find the set of values of x that satisfies the inequalities given as x squared plus 3x plus 10 greater than 0.

00:16 The second inequality is x squared minus x minus 6 is less than or equal to 0.

00:27 And the third inequality is given as 2x square plus 7x minus 9 is greater than or equal to 0.

00:38 So we have to find the set of values of x that satisfies all these inequalities, right? so first we can take this inequality over here and when we find the discriminant over here, what we find is that we get 9 minus 4 times 10 which is 40 as the discriminant which is actually less than 0 which means we can say the quadratic expression the quadratic expression is always positive for x belong to r right so there is a condition that if the discriminant is less than zero the discriminant is less than zero the quadratic expression would be having positive values for all real values of x right so that is the first set of values for x that we got when we solved x squared plus 3x plus 10 greater than 0.

02:00 In the second inequality we can first simplify this equation and write it as x squared minus 3x x square minus 3x plus 2x minus 6 is less than or equal to 0 from which we can write we can write this over here that is x square minus 3x x square minus 3x plus 2x minus 6 is less than or equal to 0 from which we can write x times x minus 3 plus 2 times x minus 3 is less than or equal to 0.

02:46 When we take x minus 3 as a common factor we will have x minus 3 times x plus 2 is less than or equal to 0.

02:54 And now we can write for this inequality to be true x should be in between x should be in between minus 2 and 3 right so for x plus 2 should be to be less than or equal to 0 x should be greater than minus 2 and for x minus 3 to be less than 0 it should be that x should have a value which is less than 3 right so in either way it should be either that x is x belongs to minus 2 x belongs to minus 2 to 3 since it is either that x is less than 3 or it is greater than minus 2 so that that is the that we get from the second equation and now the third equation what we can find is we can write 2x squared plus 7x minus 9 as we can write this as 2x square as 2x square plus 9 x minus 2x minus 9 and equal to 0 from which we can take x as a common factor from here then we would have x times 2x 2x minus 9 n 2x rather than or equal to 0 from x plus 9 minus 1 times 2x plus 9 is greater than or equal to 0.

04:28 2x plus 9 can be taken as a common factor.

04:31 Then we have 2x plus 9 times x minus 1 greater than or equal to 0...

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