뒤로Intermediate Algebra: Linear Inequalities and Interval Notation Study Guide
스터디 가이드 - 스마트 노트
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Q1. Solve the following inequality. Write the solution set using interval notation: $19 + 7x \geq 2x - 5$
Background
Topic: Linear Inequalities
This question tests your ability to solve a linear inequality and express the solution set using interval notation.
Key Terms and Formulas:
Linear inequality: An inequality involving a linear expression.
Interval notation: A way to describe the set of solutions using parentheses and brackets.
Step-by-Step Guidance
$\text{Start by isolating the variable } x \text{ on one side of the inequality.}$
$\text{Subtract } 2x \text{ from both sides to combine like terms.}$
$\text{Move constants to the other side by subtracting 19 from both sides.}$
$\text{Divide both sides by the coefficient of } x \text{ to solve for } x.$
Try solving on your own before revealing the answer!
Final Answer: $x \geq -4.8$
Interval notation: $[-4.8, \infty)$
We isolated $x$ and used interval notation to express all values greater than or equal to $-4.8$.
Q2. Solve the following inequality. Write the solution set in interval notation: $7(x - 3) < 2(4x - 1)$
Background
Topic: Linear Inequalities
This question tests your ability to solve a linear inequality with variables on both sides and write the solution in interval notation.
Key Terms and Formulas:
Distributive property: $a(b + c) = ab + ac$
Interval notation: Describes the solution set.
Step-by-Step Guidance
$\text{Apply the distributive property to both sides: } 7(x - 3) \text{ and } 2(4x - 1).$
$\text{Combine like terms to get all } x \text{ terms on one side and constants on the other.}$
$\text{Isolate } x \text{ by dividing both sides by the coefficient of } x.$
Try solving on your own before revealing the answer!
Final Answer: $x > 1.4$
Interval notation: $(1.4, \infty)$
After simplifying and isolating $x$, the solution set includes all values greater than $1.4$.
Q3. Solve the following inequality. Write the solution set in interval notation: $-3(x - 1) < -2[5 + 8(x + 5)]$
Background
Topic: Linear Inequalities with Distribution
This question tests your ability to solve inequalities involving distribution and combining like terms, then express the answer in interval notation.
Key Terms and Formulas:
Distributive property: $a(b + c) = ab + ac$
Interval notation: Describes the solution set.
Step-by-Step Guidance
$\text{Apply the distributive property to both sides.}$
$\text{Expand and simplify each side.}$
$\text{Move all } x \text{ terms to one side and constants to the other.}$
$\text{Divide by the coefficient of } x \text{ to solve for } x.$
Try solving on your own before revealing the answer!
Final Answer: $x > -2.5$
Interval notation: $(-2.5, \infty)$
After simplifying, the solution set includes all values greater than $-2.5$.
Q4. Solve: $\frac{1}{2} + \frac{3}{7} \geq \frac{x}{14}$
Background
Topic: Linear Inequalities with Fractions
This question tests your ability to solve inequalities involving fractions and express the solution in interval notation.
Key Terms and Formulas:
Fraction addition: Find a common denominator.
Interval notation: Describes the solution set.
Step-by-Step Guidance
$\text{Add the fractions on the left side by finding a common denominator.}$
$\text{Set the sum greater than or equal to } \frac{x}{14}.$
$\text{Multiply both sides by 14 to clear denominators.}$
$\text{Isolate } x \text{ and write the solution in interval notation.}$
Try solving on your own before revealing the answer!
Final Answer: $x \leq 10$
Interval notation: $(-\infty, 10]$
After combining fractions and isolating $x$, the solution set includes all values less than or equal to $10$.
Q5. Solve the following inequality. Write the solution set using interval notation: $3(x - 6) + 3x - 10 \geq 2(x - 8) + 10x$
Background
Topic: Linear Inequalities with Distribution and Combining Like Terms
This question tests your ability to solve a linear inequality with distribution and combining like terms, then express the answer in interval notation.
Key Terms and Formulas:
Distributive property: $a(b + c) = ab + ac$
Interval notation: Describes the solution set.
Step-by-Step Guidance
$\text{Apply the distributive property to both sides.}$
$\text{Combine like terms on each side.}$
$\text{Move all } x \text{ terms to one side and constants to the other.}$
$\text{Divide by the coefficient of } x \text{ to solve for } x.$
Try solving on your own before revealing the answer!
Final Answer: $x \geq 2$
Interval notation: $[2, \infty)$
After simplifying and isolating $x$, the solution set includes all values greater than or equal to $2$.
Q6. Solve the inequality: $4x + 1 < 4(x - 2)$
Background
Topic: Linear Inequalities
This question tests your ability to solve a linear inequality and express the solution in interval notation.
Key Terms and Formulas:
Distributive property: $a(b + c) = ab + ac$
Interval notation: Describes the solution set.
Step-by-Step Guidance
$\text{Apply the distributive property to the right side.}$
$\text{Move all } x \text{ terms to one side and constants to the other.}$
$\text{Solve for } x \text{ and write the solution in interval notation.}$
Try solving on your own before revealing the answer!
Final Answer: $x < -1$
Interval notation: $(-\infty, -1)$
After simplifying and isolating $x$, the solution set includes all values less than $-1$.
