Q: Solve the inequality and give the interval notation: $6x - 5 \leq 8$ or $6x - 5 \geq 13$

$(-\infty, \frac{13}{6}] \cup [3, \infty)$

Question 1

A chemist is mixing two solutions, one with a concentration of $x$ mol/L and another with a concentration of $2x$ mol/L. If the total concentration after mixing is $9$ mol/L, and the chemist used $5$ L of the first solution and $7$ L of the second, what is the value of $x$?

Accepted Answer

$x = 3.5$

Question 2

Solve for $x$: $3(7x - 5) - 6(x - 9) = 5x + 12$

Accepted Answer

$x = 8$

Question 3

A biologist is studying the growth of a bacteria population modeled by $x^2 + 13x = 40$. What are the possible values for $x$?

Accepted Answer

$x = -8, x = 5$

Question 4

Simplify the following expression: $(6 + 5i) - (8 - 6i)$

Accepted Answer

$-2 + 11i$

Question 5

A medical researcher is analyzing a quadratic equation for the spread of a virus: $x^2 + 7x + 4 = 0$. Solve by completing the square.

Accepted Answer

$x = -\frac{7}{2} \pm \frac{\sqrt{29}}{2}$

Question 6

Solve for $x$ using the quadratic formula: $x^2 - 2x + 26 = 0$

Accepted Answer

$x = 1 \pm 5i$

Question 7

A company’s profit is modeled by the equation $6x^2 = 150$. What are the possible values of $x$?

Accepted Answer

$x = \pm 5$

Question 8

Solve for $x$: $x^2 - 2x = 96$

Accepted Answer

$x = 1 \pm \sqrt{97}$

Question 9

A physicist is analyzing the motion of a particle and models its position with $x^2 + 4x - 5 = 0$. What are the solutions for $x$?

Accepted Answer

$x = -5, x = 1$

Question 10

Solve the inequality and give the interval notation: $6x - 5 \geq 8$

Accepted Answer

$x \geq \frac{13}{6}$

Question 11

A population model is given by $x^2 + 7x + 26 = 0$. What are the solutions for $x$?

Accepted Answer

$x = -\frac{7}{2} \pm \frac{\sqrt{29}}{2}i$

Question 12

Solve for $x$: $x^2 - 10x = 0$

Accepted Answer

$x = 0, x = 10$

Question 13

A business is analyzing its break-even point using the equation $4x - 3x^2 = 0$. What are the possible values for $x$?

Accepted Answer

$x = 0, x = 4$

Question 14

Solve the inequality and give the interval notation: $-55 < 3x + 13 \leq 34$

Accepted Answer

$(-6, 7]$

Question 15

Simplify: $(-4 + 2i) + (4 + 8i)$

Accepted Answer

$0 + 10i$

Question 16

Solve for $x$: $x^2 + 4x = 0$

Accepted Answer

$x = 0, x = -4$

Question 17

A physicist is analyzing a projectile’s motion and models its height with $x^2 + 7x + 26 = 0$. What is the nature of the solutions?

Accepted Answer

Two complex solutions

Question 18

Solve for $x$: $x^2 - 49 = 0$

Accepted Answer

$x = 7, x = -7$

Question 19

Solve the inequality and give the interval notation: $6x - 5 \leq 8$ or $6x - 5 \geq 13$

Accepted Answer

$(-\infty, \frac{13}{6}] \cup [3, \infty)$

Question 20

A chemical reaction is modeled by $x^2 + 35x = 51x$. What are the possible values for $x$?

Accepted Answer

$x = 0, x = 16$

College Algebra: Solving Equations, Simplifying Expressions, and Solving Inequalities

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