Algebra Interval Notation Lesson Plans & Worksheets - Free Printable

It looks like you've uploaded an image containing a math worksheet titled "Exercise Set 1.7: Interval Notation and Linear Inequalities". Since I can't directly view images, I’ll help you solve the problems if you provide the text of the questions or describe them.

However, based on the visible content in your description, here's what I can infer:

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Common Problems in This Exercise Set

This type of worksheet typically includes:
1. Solving linear inequalities.
2. Expressing solutions in interval notation.
3. Graphing solutions on a number line.
4. Determining whether certain values satisfy inequalities.

Let me go through the types of problems likely present and how to solve them.

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Example Problem Type 1: Solve and Write in Interval Notation

Problem:
Solve: $ 3x - 5 < 7 $

Solution:
$$
3x - 5 < 7 \\
3x < 12 \\
x < 4
$$

Interval Notation: $ (-\infty, 4) $

✔ Answer: $ (-\infty, 4) $

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Example Problem Type 2: Compound Inequality

Problem:
Solve: $ -2 \leq 2x + 1 < 5 $

Solution:
Break into two parts:
$$
-2 \leq 2x + 1 \quad \text{and} \quad 2x + 1 < 5
$$

Solve each:
1. $ -2 \leq 2x + 1 $ → $ -3 \leq 2x $ → $ x \geq -\frac{3}{2} $
2. $ 2x + 1 < 5 $ → $ 2x < 4 $ → $ x < 2 $

Combine: $ -\frac{3}{2} \leq x < 2 $

Interval Notation: $ \left[-\frac{3}{2}, 2\right) $

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Example Problem Type 3: Absolute Value Inequality

Problem:
Solve: $ |x - 3| \leq 2 $

Solution:
$$
-2 \leq x - 3 \leq 2 \\
1 \leq x \leq 5
$$

Interval Notation: $ [1, 5] $

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Example Problem Type 4: Which Inequality is True?

Problem:
Which of the following inequalities is true for all $ x $?
A) $ x^2 > 0 $
B) $ x + 1 > x $
C) $ x > -x $
D) $ x^2 \geq 0 $

Analysis:
- A: False when $ x = 0 $
- B: $ x + 1 > x $ → always true (subtract $ x $: $ 1 > 0 $)
- C: False if $ x < 0 $
- D: Always true (square is non-negative)

But only B and D are always true.

But if it asks “which one”, and D is correct, then D is best because it's universally true.

Wait — B is also always true.

So both B and D are true for all real $ x $. But let’s check:

- B: $ x + 1 > x $ → subtract $ x $: $ 1 > 0 $ → always true
- D: $ x^2 \geq 0 $ → always true

So both are true. But if only one answer is allowed, likely D is intended since it's more complex.

But actually, both are correct, so the question may be flawed or require choosing best.

But let’s suppose it says “which of the following must be true?” Then D is the most general.

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How to Solve Your Worksheet

To help you fully, please provide:

1. The specific problems from the image (copy-paste the text).
2. Or describe the problems (e.g., "Problem 1: Solve $ 2x + 3 \geq 7 $" etc.)

Then I can:
- Solve each step-by-step
- Give interval notation
- Explain graphing
- Show reasoning

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✔ Final Tip

For linear inequalities:
- Solve like equations, but reverse inequality when multiplying/dividing by negative number.
- Use parentheses for $ < $ or $ > $, brackets for $ \leq $ or $ \geq $.
- Graph on number line with open/closed circles.

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Please reply with the actual problems from the image, and I’ll solve them all for you!