Solve the inequality and graph the solution set.
[tex]x^2+13 x+42 \geq 0[/tex]
Answer (2)
Factor the quadratic expression: x 2 + 13 x + 42 = ( x + 6 ) ( x + 7 ) .
Find the roots: x = − 7 and x = − 6 .
Determine the intervals where the inequality holds: ( − \tinfty , − 7 ] and [ − 6 , \tinfty ) .
Express the solution set: ( − ∞ , − 7 ] ∪ [ − 6 , ∞ ) .
Explanation
Understanding the Inequality We are given the inequality x 2 + 13 x + 42 \tgeq 0 . Our goal is to find all values of x that satisfy this inequality. To do this, we will first factor the quadratic expression.
Factoring the Quadratic To factor the quadratic expression x 2 + 13 x + 42 , we look for two numbers that multiply to 42 and add to 13. These numbers are 6 and 7. Therefore, we can factor the expression as ( x + 6 ) ( x + 7 ) . So, the inequality becomes ( x + 6 ) ( x + 7 ) \tgeq 0 .
Finding Critical Points Now we need to find the intervals where ( x + 6 ) ( x + 7 ) \tgeq 0 . The roots of the quadratic equation ( x + 6 ) ( x + 7 ) = 0 are x = − 6 and x = − 7 . These roots divide the number line into three intervals: ( − \tinfty , − 7 ) , ( − 7 , − 6 ) , and ( − 6 , \tinfty ) . We will test a value from each interval to determine where the inequality holds.
Testing Intervals
Interval ( − \tinfty , − 7 ) : Let's test x = − 8 . Then ( x + 6 ) ( x + 7 ) = ( − 8 + 6 ) ( − 8 + 7 ) = ( − 2 ) ( − 1 ) = 2 . Since 2 \tgeq 0 , the inequality holds in this interval.
Interval ( − 7 , − 6 ) : Let's test x = − 6.5 . Then ( x + 6 ) ( x + 7 ) = ( − 6.5 + 6 ) ( − 6.5 + 7 ) = ( − 0.5 ) ( 0.5 ) = − 0.25 . Since − 0.25 < 0 , the inequality does not hold in this interval.
Interval ( − 6 , \tinfty ) : Let's test x = 0 . Then ( x + 6 ) ( x + 7 ) = ( 0 + 6 ) ( 0 + 7 ) = ( 6 ) ( 7 ) = 42 . Since 42 \tgeq 0 , the inequality holds in this interval.
Solution Set The inequality holds for x \tleq − 7 and x \tgeq − 6 . Therefore, the solution set is ( − \tinfty , − 7 ] \tcup [ − 6 , \tinfty ) .
Final Answer The solution set is ( − \tinfty , − 7 ] \tcup [ − 6 , \tinfty ) . This means that all values of x less than or equal to -7, and all values of x greater than or equal to -6, satisfy the inequality x 2 + 13 x + 42 \tgeq 0 .
Examples
Understanding quadratic inequalities is crucial in various real-world applications. For instance, consider a scenario where a company's profit, P ( x ) , depends on the number of units sold, x , and is modeled by a quadratic function P ( x ) = x 2 + 13 x + 42 . The company wants to determine the range of units they need to sell to ensure their profit is non-negative ( P ( x ) \tgeq 0 ). By solving the quadratic inequality x 2 + 13 x + 42 \tgeq 0 , the company can find the minimum and maximum number of units they must sell to avoid losses. This type of analysis helps businesses make informed decisions about production and sales strategies.
To solve the inequality x 2 + 13 x + 42 ≥ 0 , we factor it into ( x + 6 ) ( x + 7 ) ≥ 0 and find the roots at x = − 6 and x = − 7 . Testing intervals reveals that the solution set is ( − ∞ , − 7 ] ∪ [ − 6 , ∞ ) . Thus, the values of x that satisfy the inequality are all less than or equal to -7 or greater than or equal to -6.
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