#Linear Equations and Inequalities: Your Night-Before-the-Test Guide

Hey there! Let's get you feeling super confident about linear equations and inequalities. Think of this as your ultimate cheat sheet – everything you need, nothing you don't. Let's dive in!

#Linear Equations: The Basics

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• Form: $$ax + b = c$$, where 'a', 'b', and 'c' are real numbers and $$a ≠ 0$$. It's all about that straight line!
• Goal: Isolate the variable (usually 'x') on one side of the equation.
• Solution: The value of 'x' that makes the equation true.

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• "Undo" Operations: Use opposite operations to isolate 'x'.
  • If it's addition, subtract. If it's multiplication, divide.
  • Remember: Whatever you do to one side, you must do to the other!
• Simplify: Combine like terms and use the distributive property (if needed) to clean things up.
  • Distributive property: $$a(b+c) = ab + ac$$

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• One Solution: A single value of 'x' works. Example: $$2x + 3 = 7$$ (x = 2)

• No Solution (Inconsistent): You end up with a false statement, like $$3 = 5$$. Example: $$2x + 5 = 2x + 8$$.

• Infinitely Many Solutions (Identity): The equation is true for any value of 'x'. Example: $$2x + 4 = 2(x + 2)$$.

• Verification: Always plug your solution back into the original equation to check your work.

Practice Question

Multiple Choice Questions:

1. Solve for x: $$3x - 7 = 14$$ a) 3 b) 7 c) 21/3 d) 21

2. Which of the following equations has no solution? a) $$2x + 5 = 11$$ b) $$3x - 4 = 3x + 2$$ c) $$4x + 8 = 4(x + 2)$$ d) $$5x - 10 = 0$$

Free Response Question:

Solve the following equation for x and show all your steps: $$5(2x - 3) + 10 = 3x + 15$$

Scoring Rubric:

• 1 point: Distribute the 5 correctly: $$10x - 15 + 10 = 3x + 15$$
• 1 point: Combine like terms: $$10x - 5 = 3x + 15$$
• 1 point: Subtract 3x from both sides: $$7x - 5 = 15$$
• 1 point: Add 5 to both sides: $$7x = 20$$
• 1 point: Divide by 7: $$x = 20/7$$

#Linear Inequalities: Stepping Up

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• Form: $$ax + b > c$$, $$ax + b ≥ c$$, $$ax + b < c$$, or $$ax + b ≤ c$$. Similar to equations, but with a range of solutions.
• Goal: Isolate the variable (usually 'x') on one side of the inequality.
• Key Difference: When you multiply or divide by a negative number, you must flip the inequality sign. 💡

#Representing Solutions: Intervals, Sets, and Graphs

• Solution Set: All values of 'x' that make the inequality true.
• Interval Notation: - Parentheses () for strict inequalities (<$ or>$$). Example:$$(2, 5)$$means all numbers between 2 and 5, not including 2 and 5. - Brackets [object Object] for inclusive inequalities ($$\leq$$or$$\geq$$). Example:$$[2, 5]$$means all numbers between 2 and 5, including 2 and 5. - [object Object] {x | condition}. Example: {x | x > 3} means all 'x' such that 'x' is greater than 3. - [object Object] - Open circles [object Object] for strict inequalities ($$<$or$$>$$). - Closed circles [object Object] for inclusive inequalities ($$\leq$$or$$\geq$$).$

#Solving Techniques: Same as Equations, with a Twist

• Addition/Subtraction: Add or subtract the same value from both sides (no sign change).
• Multiplication/Division: - Positive numbers: Multiply or divide without changing the inequality sign. - Negative numbers: Multiply or divide and flip the inequality sign.
• Simplify: Combine like terms and use the distributive property when necessary.

<practice_question>

Multiple Choice Questions:

1. Solve for x:$-2x + 5 < 11$$a)$$x < -3$$b)$$x > -3$$c)$$x < 8$$d)$$x > 8$$2. Which interval notation represents the inequality$$x \geq -2$$? a)$$(-2, \infty)$$b)$$[-2, \infty)$$c)$$(-\infty, -2)$$d)$$(-\infty, -2]$$[object Object]$

Solve the following inequality for x and show all your steps. Represent your answer in interval notation and on a number line:$3(x - 2) + 4 \geq 2x - 1$$[object Object]$

• 1 point: Distribute the 3 correctly:$3x - 6 + 4 \geq 2x - 1$$- 1 point: Combine like terms:$$3x - 2 \geq 2x - 1$$- 1 point: Subtract 2x from both sides:$$x - 2 \geq -1$$- 1 point: Add 2 to both sides:$$x \geq 1$$- 1 point: Correct interval notation:$$[1, \infty)$$- 1 point: Correct number line representation (closed circle at 1, arrow pointing to the right).$

</practice_question>

#<high_value_topic> Solutions in Context: Real-World Applications </high_value_topic>

#Interpreting Equation Solutions: What Does It Mean?

• Specific Values: Solutions represent specific values that satisfy the conditions of the problem.
• Context is Key: Always relate your mathematical answer back to the original problem.
• Units Matter: Include units in your answer (e.g., dollars, hours, meters).
• Feasibility: Does the solution make sense in the real world? (e.g., can't have negative time).

#Interpreting Inequality Solutions: A Range of Possibilities

• Ranges of Values: Solutions represent a range of values that satisfy the problem's conditions.
• Upper/Lower Bounds: Understand the limits of your solution range.
• Realistic Values: Make sure all values within the solution set are realistic for the situation.

#Applying Solutions to Real-World Problems: Making Math Useful

• Modeling: Use equations and inequalities to represent real-life situations.
• Constraints: Consider any limitations or restrictions given in the problem.
• Clear Explanations: Communicate your solutions with units and clear explanations.
• Examples: If 'x' represents time, explain what a solution of$x > 5$$means in terms of hours or minutes.$

<practice_question>

Multiple Choice Questions:

1. A taxi charges a </math-inline>3 initial fee plus <math-inline>2 per mile. If a ride costs </math-inline>15, how many miles was the ride? a) 5 miles b) 6 miles c) 7 miles d) 8 miles

2. A student needs at least 80 points to get a B in a class. If they have 65 points now, and each assignment is worth 5 points, how many assignments do they need to complete? a) 2 b) 3 c) 4 d) 5

Free Response Question:

A company produces widgets. The cost to produce each widget is <math-inline>5 and the company has fixed costs of </math-inline>100. The company sells each widget for $12. Write an inequality to determine how many widgets the company needs to sell to make a profit. Solve the inequality and explain your answer in the context of the problem.

Scoring Rubric:

• 1 point: Define variables: Let x be the number of widgets.
• 1 point: Write the cost equation: Cost = 5x + 100
• 1 point: Write the revenue equation: Revenue = 12x
• 1 point: Write the profit inequality: 12x > 5x + 100
• 1 point: Solve the inequality: 7x > 100, x > 100/7, x > 14.29
• 1 point: Explain in context: The company needs to sell at least 15 widgets to make a profit (since you can't sell a fraction of a widget).

#Final Exam Focus: Last-Minute Tips

• High-Priority Topics: Linear equations, linear inequalities, and their real-world applications.
• Common Question Types: Solving equations and inequalities, interpreting solutions in context, word problems.

- - **Strategies:** - Read each question carefully and underline key information. - Show all your work, even for multiple-choice questions. - Check your answers by plugging them back into the original equation or inequality.

You've got this! Go into the exam with confidence. You're well-prepared and ready to rock! 🚀