Precalculus Study Guide: Interval Notation & Solving Inequalities

Q1. Complete the table: Express the interval $x < 5$ in interval notation, inequality, set notation, and graph it on the number line.

Background

Topic: Interval Notation & Representing Inequalities

This question tests your ability to translate between different ways of expressing intervals: interval notation, inequalities, set notation, and graphical representation.

Key Terms:

• Interval Notation: Uses parentheses and brackets to describe sets of numbers.

• Inequality: Mathematical statement comparing values.

• Set Notation: Describes a set using curly braces and conditions.

• Graph: Visual representation on the number line.

Step-by-Step Guidance

1. Start by identifying the inequality: $x < 5$.

2. Translate this into interval notation. Recall that values less than 5 extend to negative infinity.

3. Write the set notation: Use curly braces and a condition for $x$.

4. For the graph, mark all values less than 5 on the number line, using an open circle at 5.

Try solving on your own before revealing the answer!

Final Answer:

• Interval Notation: $(-\infty, 5)$

• Inequality: $x < 5$

• Set Notation: $\{x \mid x < 5\}$

• Graph: Shade all values to the left of 5, open circle at 5.

This shows all numbers less than 5, not including 5 itself.

Q2. Complete the table: Express the interval $x < 1$ in interval notation, inequality, set notation, and graph it on the number line.

Background

Topic: Interval Notation & Representing Inequalities

This question is similar to Q1, but with a different boundary value.

Key Terms:

• Interval Notation

• Inequality

• Set Notation

• Graph

Step-by-Step Guidance

1. Start with the inequality: $x < 1$.

2. Express this in interval notation, considering values less than 1.

3. Write the set notation for $x$ values less than 1.

4. On the number line, shade all values to the left of 1, open circle at 1.

Try solving on your own before revealing the answer!

Final Answer:

• Interval Notation: $(-\infty, 1)$

• Inequality: $x < 1$

• Set Notation: $\{x \mid x < 1\}$

• Graph: Shade all values to the left of 1, open circle at 1.

Q3. Complete the table: Express the interval $x \geq 1$ in interval notation, inequality, set notation, and graph it on the number line.

Background

Topic: Interval Notation & Representing Inequalities

This question tests your understanding of intervals that include the boundary value.

Key Terms:

• Interval Notation

• Inequality

• Set Notation

• Graph

Step-by-Step Guidance

1. Start with the inequality: $x \geq 1$.

2. Express this in interval notation, remembering to include 1.

3. Write the set notation for $x$ values greater than or equal to 1.

4. On the number line, shade all values to the right of 1, closed circle at 1.

Try solving on your own before revealing the answer!

Final Answer:

• Interval Notation: $[1, \infty)$

• Inequality: $x \geq 1$

• Set Notation: $\{x \mid x \geq 1\}$

• Graph: Shade all values to the right of 1, closed circle at 1.

Q4. Complete the table: Express the interval $1 \leq x < 3$ in interval notation, inequality, set notation, and graph it on the number line.

Background

Topic: Compound Inequalities & Interval Notation

This question tests your ability to represent a range of values between two endpoints, including one and excluding the other.

Key Terms:

• Compound Inequality

• Interval Notation

• Set Notation

• Graph

Step-by-Step Guidance

1. Start with the compound inequality: $1 \leq x < 3$.

2. Express this in interval notation, noting which endpoints are included/excluded.

3. Write the set notation for $x$ values between 1 and 3.

4. On the number line, shade between 1 and 3, closed circle at 1, open circle at 3.

Try solving on your own before revealing the answer!

Final Answer:

• Interval Notation: $[1, 3)$

• Inequality: $1 \leq x < 3$

• Set Notation: $\{x \mid 1 \leq x < 3\}$

• Graph: Shade between 1 and 3, closed at 1, open at 3.

Q5. Complete the table: Express the interval $[2, 4)$ in inequality, set notation, and graph it on the number line.

Background

Topic: Interval Notation & Representing Intervals

This question tests your ability to convert interval notation to other forms.

Key Terms:

• Interval Notation

• Inequality

• Set Notation

• Graph

Step-by-Step Guidance

1. Start with interval notation: $[2, 4)$.

2. Translate to inequality, noting which endpoints are included/excluded.

3. Write the set notation for $x$ values between 2 and 4.

4. On the number line, shade between 2 and 4, closed at 2, open at 4.

Try solving on your own before revealing the answer!

Final Answer:

• Inequality: $2 \leq x < 4$

• Set Notation: $\{x \mid 2 \leq x < 4\}$

• Graph: Shade between 2 and 4, closed at 2, open at 4.

Q6. Complete the table: Express the interval $[2, \infty)$ in inequality, set notation, and graph it on the number line.

Background

Topic: Interval Notation & Representing Intervals

This question tests your ability to represent intervals that extend infinitely in one direction.

Key Terms:

• Interval Notation

• Inequality

• Set Notation

• Graph

Step-by-Step Guidance

1. Start with interval notation: $[2, \infty)$.

2. Translate to inequality, noting the inclusion of 2 and extension to infinity.

3. Write the set notation for $x$ values greater than or equal to 2.

4. On the number line, shade all values to the right of 2, closed at 2.

Try solving on your own before revealing the answer!

Final Answer:

• Inequality: $x \geq 2$

• Set Notation: $\{x \mid x \geq 2\}$

• Graph: Shade all values to the right of 2, closed at 2.

Q7. Complete the table: Express the interval $(-\infty, 0)$ in inequality, set notation, and graph it on the number line.

Background

Topic: Interval Notation & Representing Intervals

This question tests your ability to represent intervals that extend infinitely in the negative direction.

Key Terms:

• Interval Notation

• Inequality

• Set Notation

• Graph

Step-by-Step Guidance

1. Start with interval notation: $(-\infty, 0)$.

2. Translate to inequality, noting the exclusion of 0.

3. Write the set notation for $x$ values less than 0.

4. On the number line, shade all values to the left of 0, open at 0.

Try solving on your own before revealing the answer!

Final Answer:

• Inequality: $x < 0$

• Set Notation: $\{x \mid x < 0\}$

• Graph: Shade all values to the left of 0, open at 0.

Q8. Complete the table: Express the interval $(-\infty, 0]$ in inequality, set notation, and graph it on the number line.

Background

Topic: Interval Notation & Representing Intervals

This question tests your ability to represent intervals that include a boundary value and extend infinitely.

Key Terms:

• Interval Notation

• Inequality

• Set Notation

• Graph

Step-by-Step Guidance

1. Start with interval notation: $(-\infty, 0]$.

2. Translate to inequality, noting the inclusion of 0.

3. Write the set notation for $x$ values less than or equal to 0.

4. On the number line, shade all values to the left of 0, closed at 0.

Try solving on your own before revealing the answer!

Final Answer:

• Inequality: $x \leq 0$

• Set Notation: $\{x \mid x \leq 0\}$

• Graph: Shade all values to the left of 0, closed at 0.

Q9. Real-world Application: Fruits cannot be marketed as “organic” unless the soil has been pesticide-free for at least three years. Let $t$ be the number of years the soil has been pesticide-free. Describe the statement in interval notation, inequality, set notation, and graph it on the number line.

Background

Topic: Translating Real-World Conditions to Mathematical Language

This question tests your ability to model a real-world scenario using interval notation, inequalities, set notation, and graphs.

Key Terms:

• Interval Notation

• Inequality

• Set Notation

• Graph

Step-by-Step Guidance

1. Identify the condition: $t$ must be at least 3 years.

2. Express this as an inequality, considering "at least" means greater than or equal to.

3. Write the interval notation for values of $t$ starting at 3 and extending to infinity.

4. Write the set notation for $t$ values meeting the condition.

5. On the number line, shade all values to the right of 3, closed at 3.

Try solving on your own before revealing the answer!

Final Answer:

• Interval Notation: $[3, \infty)$

• Inequality: $t \geq 3$

• Set Notation: $\{t \mid t \geq 3\}$

• Graph: Shade all values to the right of 3, closed at 3.

This models the requirement that the soil must be pesticide-free for at least 3 years.

Q10. Real-world Application: Did you use one of $\leq$ or $\geq$, or one of $<$ or $>$? Why?

Background

Topic: Interpreting Inequality Symbols in Context

This question tests your understanding of how to choose the correct inequality symbol based on the wording of a real-world scenario.

Key Terms:

• Inequality Symbols: $<$, $>$, $\leq$, $\geq$

• "At least" means including the boundary value.

Step-by-Step Guidance

1. Recall the phrase "at least three years" from the scenario.

2. Determine whether the boundary value (3 years) should be included.

3. Choose the appropriate inequality symbol to reflect inclusion.

4. Explain your reasoning for the choice.

Try solving on your own before revealing the answer!

Final Answer:

You should use $\geq$ (greater than or equal to) because "at least" means the value 3 is included. If you used $>$, it would exclude 3, which does not match the requirement.

Q11. Shade the interval(s) $[−2, 1) \cup (2, \infty)$ on the number line.

Background

Topic: Union of Intervals & Graphing

This question tests your ability to graph multiple intervals and understand union notation.

Key Terms:

• Union ($\cup$): Combines two or more intervals.

• Interval Notation

• Graph

Step-by-Step Guidance

1. Identify the two intervals: $[−2, 1)$ and $(2, \infty)$.

2. For $[−2, 1)$, shade from $-2$ (closed) up to $1$ (open).

3. For $(2, \infty)$, shade all values greater than 2, open at 2.

4. On the number line, represent both intervals, using appropriate open/closed circles.

Try solving on your own before revealing the answer!

Final Answer:

Shade from $-2$ (closed) to $1$(open), and from $2$ (open) to infinity. Leave gaps between $1$and $2$.

Q12. Shade the interval(s) $(−\infty, 1) \cup (2, 3) \cup (6, \infty)$ on the number line.

Background

Topic: Union of Multiple Intervals & Graphing

This question tests your ability to graph several intervals and understand union notation.

Key Terms:

• Union ($\cup$)

• Interval Notation

• Graph

Step-by-Step Guidance

1. Identify the three intervals: $(−\infty, 1)$, $(2, 3)$, and $(6, \infty)$.

2. For $(−\infty, 1)$, shade all values less than 1, open at 1.

3. For $(2, 3)$, shade between 2 and 3, open at both ends.

4. For $(6, \infty)$, shade all values greater than 6, open at 6.

5. On the number line, represent all three intervals, leaving gaps between them.

Try solving on your own before revealing the answer!

Final Answer:

Shade all values less than 1 (open at 1), between 2 and 3 (open at both ends), and greater than 6 (open at 6).

Q13. Solve the inequality $5x - 9 < 1$ and state your answer using interval notation.

Background

Topic: Solving Linear Inequalities

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

Key Terms and Formulas:

• Linear Inequality: An inequality involving a linear expression.

• Interval Notation

Step-by-Step Guidance

1. Start with the inequality: $5x - 9 < 1$.

2. Add 9 to both sides to isolate the $5x$ term.

3. Divide both sides by 5 to solve for $x$.

4. Express the solution as an inequality and then in interval notation.

Try solving on your own before revealing the answer!

Final Answer:

$x < 2$

Interval Notation: $(-\infty, 2)$

Q14. Solve the inequality $-5x + 1 \leq 2$ and state your answer using interval notation.

Background

Topic: Solving Linear Inequalities

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

Key Terms and Formulas:

• Linear Inequality

• Interval Notation

Step-by-Step Guidance

1. Start with the inequality: $-5x + 1 \leq 2$.

2. Subtract 1 from both sides to isolate the $-5x$ term.

3. Divide both sides by $-5$ (remember to reverse the inequality sign when dividing by a negative).

4. Express the solution as an inequality and then in interval notation.

Try solving on your own before revealing the answer!

Final Answer:

$x \geq -0.2$

Interval Notation: $[-0.2, \infty)$

Remember, dividing by a negative reverses the inequality sign.

Q15. Solve the inequality $\frac{1}{3}(x - 9) > 2$ and state your answer using interval notation.

Background

Topic: Solving 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:

• Linear Inequality

• Interval Notation

Step-by-Step Guidance

1. Start with the inequality: $\frac{1}{3}(x - 9) > 2$.

2. Multiply both sides by 3 to eliminate the fraction.

3. Solve for $x$ by isolating it on one side.

4. Express the solution as an inequality and then in interval notation.

Try solving on your own before revealing the answer!

Final Answer:

$x > 15$

Interval Notation: $(15, \infty)$

Q16. Solve the inequality $3 + 7x + 5 \geq 17$ and state your answer using interval notation.

Background

Topic: Solving Linear Inequalities

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

Key Terms and Formulas:

• Linear Inequality

• Interval Notation

Step-by-Step Guidance

1. Start with the inequality: $3 + 7x + 5 \geq 17$.

2. Combine like terms on the left side.

3. Subtract the constant from both sides to isolate the $7x$ term.

4. Divide both sides by 7 to solve for $x$.

5. Express the solution as an inequality and then in interval notation.

Try solving on your own before revealing the answer!

Final Answer:

$x \geq 1.27$

Interval Notation: $[1.27, \infty)$