Lesson 2.1: Linear Equations and Inequalities

Introduction

In this lesson, we will explore the foundational concepts of linear equations and inequalities. Mastering these concepts is crucial, as they form the basis for more advanced mathematical topics. Our objectives for this lesson are to:

Solve linear equations, including those with brackets and fractions.
Solve linear inequalities and represent solutions on a number line.
Set up an equation or inequality from a worded problem.
Solve a linear equation involving brackets and fractions.
Solve a linear inequality and represent the solution as an interval.

By the end of this lesson, you, students, will feel confident in tackling linear equations and inequalities and applying them to real-world situations.

Section 1: Understanding Linear Equations

What is a Linear Equation?

A linear equation is an algebraic equation of the form:

$ax + b = c$

where:

$x$ is the variable we want to solve for,
$a$ and $b$ are constants, and
$c$ is a constant.

Linear equations are called "linear" because their graph is a straight line when plotted on a coordinate plane. The solutions to linear equations can be found by isolating the variable on one side of the equation.

Example 1: Solving a Simple Linear Equation

Let’s solve the equation:

$$ 2x + 3 = 11 $$

To solve for $x$, we follow these steps:

Subtract 3 from both sides:

$$ 2x + 3 - 3 = 11 - 3 $$

$$ 2x = 8 $$

Divide by 2:

$$ x = \frac{8}{2} $$

$$ x = 4 $$

So, the solution to the equation $2x + 3 = 11$ is $x = 4$.

Solving Linear Equations with Brackets

Sometimes, linear equations may include brackets. It's important to apply the distributive property to remove the brackets first.

Example 2: Solving an Equation with Brackets

Solve the equation:

$$ 3(2x - 1) = 9 $$

Step 1: Distribute $3$ on the left side:

$$ 6x - 3 = 9 $$

Step 2: Add $3$ to both sides:

$$ 6x = 12 $$

Step 3: Divide by $6$:

$$ x = 2 $$

Therefore, the solution is $x = 2$.

Solving Linear Equations with Fractions

When an equation contains fractions, it is often helpful to eliminate the fractions by multiplying both sides by the least common denominator (LCD).

Example 3: Solving an Equation with Fractions

Solve:

$$ \frac{x}{4} + 3 = 7 $$

Step 1: Subtract $3$ from both sides:

$$ \frac{x}{4} = 4 $$

Step 2: Multiply both sides by $4$:

$$ x = 16 $$

Thus, the solution is $x = 16$.

Section 2: Linear Inequalities

What is a Linear Inequality?

A linear inequality is similar to a linear equation but uses inequality symbols instead of an equals sign. The general form is:

$$ ax + b < c $$

$$ ax + b > c $$

Example 4: Solving a Linear Inequality

Solve the inequality:

$$ 3x - 4 > 5 $$

Step 1: Add $4$ to both sides:

$$ 3x > 9 $$

Step 2: Divide by $3$:

$$ x > 3 $$

Representing Solutions on a Number Line

To represent the solution $x > 3$ on a number line, you would draw a circle (not filled in) at $3$ and shade the line extending to the right, indicating all numbers greater than $3$.

Example 5: Solving a Linear Inequality with Brackets

Solve:

$$ 2(3x + 1) \leq 10 $$

Step 1: Distribute $2$:

$$ 6x + 2 \leq 10 $$

Step 2: Subtract $2$ from both sides:

$$ 6x \leq 8 $$

Step 3: Divide by $6$:

$$ x \leq \frac{8}{6} $$

$$ x \leq \frac{4}{3} $$

Representing $x \leq \frac{4}{3}$ on a number line: circle at $ \frac{4}{3} $ filled in, shaded to the left.

Section 3: Setting Up Linear Equations and Inequalities from Word Problems

Example 6: Word Problem to Equation

A car rental company charges a flat fee of $50 plus $0.20 per mile driven. If a customer pays $70, how many miles did they drive?

Step 1: Set up the equation:

$$ 50 + 0.20m = 70 $$

Step 2: Subtract $50$:

$$ 0.20m = 20 $$

Step 3: Divide by $0.20$:

$$ m = 100 $$

Thus, the customer drove $100$ miles.

Example 7: Word Problem to Inequality

A bank offers a savings account with an annual interest rate of $3\% $ on deposits. If a customer wants to earn at least $15 in interest in a year, how much money do they need to deposit?

Step 1: Set up the inequality:

$$ 0.03d \geq 15 $$

Step 2: Divide by $0.03$:

$$ d \geq 500 $$

So, the customer must deposit at least $500.

Conclusion

In this lesson, we covered the essential techniques for solving linear equations and inequalities. You learned how to manipulate equations with both brackets and fractions, represented solutions on a number line, and translated word problems into mathematical equations and inequalities. Mastery of these concepts will greatly enhance your ability to tackle more complex mathematical problems.

Study Notes

A linear equation has the form $ ax + b = c $.
A linear inequality uses symbols like <, >, ≤, or ≥.
To solve equations with brackets, apply the distributive property.
To eliminate fractions, multiply by the least common denominator.
Solutions of inequalities can be represented on a number line.
Translate word problems into equations or inequalities for solving.