Answer:
![x < - 6 \ \text{and} \ x > 1](https://img.qammunity.org/2023/formulas/mathematics/college/tqeamz0cj212zkl6qt5wcdn8n0es79ot7f.png)
Explanation:
Givens
We are given an inequality:
![7x^2 + 35x > 42](https://img.qammunity.org/2023/formulas/mathematics/college/rftmlp8brzwq0cst6s9rtkjglgij3urw9e.png)
We are asked to find the solution to this inequality.
Solve
First, treat this as a quadratic equation. Get the inequality to quadratic form:
![ax^2+bx+c > 0\\\\7x^2+35x > 42\\\\7x^2+35x-42 > 42-42\\\\7x^2+35x-42 > 0](https://img.qammunity.org/2023/formulas/mathematics/college/ei1b79ifb2ao6byxhpu1fk481nh3itc250.png)
Then, divide both sides by 7:
![\displaystyle 7x^2+35x-42 > 0\\\\(7x^2+35x-42)/(7) > (0)/(7)\\\\x^2+5x-6 > 0](https://img.qammunity.org/2023/formulas/mathematics/college/pxnxzu74gt78aw6fwbrlkxpkdxtri3u3cx.png)
Then, separate 5x to create a factorable expression. Ask yourself: what two numbers will add up to -6 and multiply to create -6?
![x^2+5x-6 > 0\\\\x^2+6x-x-6 > 0](https://img.qammunity.org/2023/formulas/mathematics/college/qv855tiepal3fqpvw4vx9iavdbu2kx77h2.png)
Then, group the terms together:
![x^2+6x-x-6 > 0\\\\(x^2+6x)(-x-6) > 0](https://img.qammunity.org/2023/formulas/mathematics/college/eed089wb8lo9im2y6syyrfcy9agjw0tv58.png)
Then, factor out x from the inequality:
![(x^2+6x)(-x-6) > 0\\\\x(x+6)(-x-6) > 0](https://img.qammunity.org/2023/formulas/mathematics/college/trd4a44ic8z70nteshc086d1t9r9i07czq.png)
Then, factor out the negative sign from the second part of the expression:
![x(x+6)(-x-6) > 0\\\\x(x+6)-(x+6) > 0](https://img.qammunity.org/2023/formulas/mathematics/college/syw94o2kfmmzj2bzm3wan855iq9jk11xct.png)
Then, factor out x + 6 from the inequality:
![x(x+6)-(x+6) > 0\\\\(x+6)(x-1) > 0](https://img.qammunity.org/2023/formulas/mathematics/college/1eqrl2lnya99th6mpsapwyf1mhnnp7spg7.png)
Then, separate the factors and create two separate inequality sets:
![(x+6)(x-1) > 0\\\\\displaystyle \left \{ {{x+6 > 0} \atop {x-1 > 0}} \right. \\\\\left \{ {{x+6 < 0} \atop {x-1 < 0}} \right.](https://img.qammunity.org/2023/formulas/mathematics/college/dat1rk2c6w58iscw89w9bx6d23jn1sb53j.png)
Then, solve the inequalities for x:
![x+6 > 0\\\\x+6-6 > 0-6\\\\x > -6\\---------------\\x-1 > 0\\\\x-1+1 > 0+1\\\\x > 1\\---------------\\x+6 < 0\\\\x+6-6 < 0-6\\\\x < -6\\---------------\\x-1 < 0\\\\x-1+1 < 0+1\\\\x < 1](https://img.qammunity.org/2023/formulas/mathematics/college/tjp9kdpasmciux78z5gu5uigkkwb5ppqrf.png)
Then, test each value in the inequality:
![7x^2+35x > 42\\\\7(-5)^2+35(-5) > 42\\\\7(25)-175 > 42\\\\175-175 > 42\\\\0 \\gtr 42\\\\\text{x} > \text{-6 is not a solution.}\\---------------\\7x^2+35x > 42\\\\7(2)^2+35(2) > 42\\\\7(4)+70 > 42\\\\28+70 > 42\\\\98 > 42\\\\\text{x} > \text{1 is a solution.}\\---------------](https://img.qammunity.org/2023/formulas/mathematics/college/r241oyxdato5o16joqu400jibkrewuz7s3.png)
![\\7x^2+35x > 42\\\\7(-7)^2+35(-7) > 42\\\\7(49)-245 > 42\\\\343-245 > 42\\\\98 > 42\\\\\text{x} < \text{-6 is a solution.}\\---------------\\7x^2+35x > 42\\\\7(-1)^2+35(-1) > 42\\\\7(1)-35 > 42\\\\7-35 > 42\\\\-28\\gtr42\\\\\text{x} < \text{1 is not a solution.}](https://img.qammunity.org/2023/formulas/mathematics/college/3f3qcbysrlxsj2b7gpn0t58ylm1owy3vu6.png)
Final Answer
Therefore, the solutions are x > 1 and x < -6. This means that answer choice A is correct.