College Algebra Exam Review: Step-by-Step Guidance

Q1a. Solve the equation: $9x + 11 = 7x + 1$

Background

Topic: Linear Equations

This question tests your ability to solve a basic linear equation for the variable $x$.

Key Terms and Formulas:

• Linear equation: An equation of the form $ax + b = cx + d$.

• Solving for $x$: Isolate $x$ on one side of the equation.

Step-by-Step Guidance

1. Subtract $7x$ from both sides to get all $x$ terms on one side: $9x + 11 - 7x = 7x + 1 - 7x$.

2. Simplify both sides: $2x + 11 = 1$.

3. Subtract $11$from both sides to isolate$x$: $2x = 1 - 11$.

Try solving on your own before revealing the answer!

Final Answer: $x = -5$

Subtracting $11$gives $2x = -10$, then dividing by $2$yields$x = -5$.

Q2a. Simplify: $\sqrt{-90}$

Background

Topic: Complex Numbers

This question tests your understanding of imaginary numbers and how to write square roots of negative numbers in standard form.

Key Terms and Formulas:

• Imaginary unit: $i$, where $i^2 = -1$.

• Standard form: $a + bi$.

• $\sqrt{-a} = i\sqrt{a}$ for $a > 0$.

Step-by-Step Guidance

1. Recognize that $\sqrt{-90}$ can be rewritten as $i\sqrt{90}$.

2. Factor $90$to see if it can be simplified: $90 = 9 \times 10$.

3. Use $\sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10}$.

4. So $\sqrt{90} = 3\sqrt{10}$, thus $\sqrt{-90} = 3i\sqrt{10}$.

Try solving on your own before revealing the answer!

Final Answer: $3i\sqrt{10}$

We simplified $\sqrt{-90}$ using the imaginary unit and factoring $90$.

Q4a. Solve by factoring: $x^2 + 2x - 35 = 0$

Background

Topic: Quadratic Equations

This question tests your ability to solve a quadratic equation by factoring.

Key Terms and Formulas:

• Quadratic equation: $ax^2 + bx + c = 0$.

• Factoring: Expressing the quadratic as a product of two binomials.

Step-by-Step Guidance

1. Identify $a = 1$, $b = 2$, $c = -35$.

2. Look for two numbers that multiply to $-35$ and add to $2$.

3. Write the equation as $(x + m)(x + n) = 0$, where $m$ and $n$ are the numbers found.

4. Set each factor equal to zero: $x + m = 0$ and $x + n = 0$.

Try solving on your own before revealing the answer!

Final Answer: $x = 5$ or $x = -7$

The numbers $m$$and$ are $5$ and $-7$, so the solutions are$x = 5$ and $x = -7$.

Q8. A square flower bed is to be enlarged by adding 2 meters on each side. If the larger square has an area of 196 square meters, what is the length of a side of the original square?

Background

Topic: Applications and Modeling with Quadratic Equations

This question tests your ability to set up and solve a quadratic equation based on a real-world scenario involving area.

Key Terms and Formulas:

• Area of a square: $A = s^2$, where $s$ is the side length.

• Enlarged side: Original side $+ 2$ meters.

Step-by-Step Guidance

1. Let $x$ be the length of the original side.

2. The enlarged side is $x + 2$ meters.

3. The area of the enlarged square is $(x + 2)^2 = 196$.

4. Expand $(x + 2)^2$ to get $x^2 + 4x + 4 = 196$.

5. Subtract $196$from both sides to set the equation to zero:$x^2 + 4x + 4 - 196 = 0$.

Try solving on your own before revealing the answer!

Final Answer: $x = 12$ meters

Solving the quadratic equation gives the original side length as $12$ meters.

Q13a. Solve the inequality: $-5x - 4 \geq 3(2x - 5)$

Background

Topic: Linear Inequalities

This question tests your ability to solve linear inequalities and express the solution in interval notation.

Key Terms and Formulas:

• Linear inequality: An inequality involving a linear expression.

• Interval notation: A way to express the set of solutions.

Step-by-Step Guidance

1. Expand the right side: $3(2x - 5) = 6x - 15$.

2. Rewrite the inequality: $-5x - 4 \geq 6x - 15$.

3. Add $5x$ to both sides: $-4 \geq 11x - 15$.

4. Add $15$to both sides: $11 \geq 11x$.

5. Divide both sides by $11$to isolate$x$.

Try solving on your own before revealing the answer!

Final Answer: $x \leq 1$

The solution in interval notation is $(-\infty, 1]$.