BackCollege Algebra Study Guidance: Solving Inequalities, Graphing Solutions, and Simplifying Expressions
Study Guide - Smart Notes
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Q9. Solve: $5x + 2 \geq 12$
Background
Topic: Linear Inequalities
This question tests your ability to solve linear inequalities and interpret the solution in terms of the variable $x$.
Key Terms and Formulas:
Linear inequality: An inequality involving a linear expression, such as $ax + b \geq c$.
Solving inequalities: Similar to solving equations, but pay attention to the direction of the inequality sign.
Step-by-Step Guidance
$\text{Start by isolating the variable } x \text{ on one side of the inequality.}$
$\text{Subtract 2 from both sides: } 5x + 2 - 2 \geq 12 - 2$
$\text{Simplify: } 5x \geq 10$
$\text{Divide both sides by 5 to solve for } x.$
Try solving on your own before revealing the answer!
Final Answer: $x \geq 2$
$\text{Dividing both sides by 5 gives } x \geq 2.$
This means all values of $x$ greater than or equal to 2 satisfy the inequality.
Q10. Which answer choice below is the graph of $x \leq 3$?
Background
Topic: Graphing Solutions to Inequalities
This question tests your ability to interpret and graph the solution set of an inequality on a number line.
Key Terms and Concepts:
Number line: A visual representation of all real numbers.
Closed circle: Indicates the endpoint is included (e.g., $x \leq 3$ means 3 is included).
Arrow: Shows the direction of the solution set (left for $\leq$, right for $\geq$).
Step-by-Step Guidance
$\text{Identify the inequality: } x \leq 3$
$\text{On a number line, this means all values to the left of 3, including 3 itself.}$
$\text{Look for a graph with a closed circle at 3 and an arrow pointing left.}$
Try solving on your own before revealing the answer!
Final Answer: Choice a
The correct graph is the one with a closed circle at 3 and an arrow extending to the left, representing all values less than or equal to 3.
Q11. Simplify: $3x + x + 5x + 2x + x$
Background
Topic: Combining Like Terms
This question tests your ability to simplify algebraic expressions by combining like terms.
Key Terms and Concepts:
Like terms: Terms that have the same variable raised to the same power (e.g., $x$ terms).
Simplifying: Adding or subtracting coefficients of like terms.
Step-by-Step Guidance
$\text{List all the coefficients of } x: 3, 1, 5, 2, 1$
$\text{Add the coefficients together: } 3 + 1 + 5 + 2 + 1$
$\text{Write the simplified expression as } (\text{sum of coefficients})x$
Try solving on your own before revealing the answer!
Final Answer: $12x$
Adding all the coefficients gives $12x$ as the simplified expression.
