Solving linear equations and linear inequalities

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Duration: 4:32

Note: If you're taking the SAT, then chances are you have a good understanding of how to solve linear equations and inequalities. However, it's important to solve them in their various forms with consistency. We recommend that you write out your steps (instead of doing everything in your head) to avoid careless errors, and we will do the same in our examples!

You can learn anything. Let's do this!

What are linear equations and inequalities?

Linear equations and inequalities are composed of constants and variables.

Linear equations use the equal sign (=)
Linear inequalities use inequality signs (>, <, ≥, ≤)

In this lesson, we'll learn to:

Solve linear equations
Solve linear inequalities
Recognize conditions for one solution, no solution, or infinite solutions

How do I solve linear equations?

Reasoning with linear equations

The goal is to find the value of the variable by isolating it step by step until only the variable is on one side and a constant is on the other.

Key rule: The equation remains equivalent only if we treat both sides equally - whatever we do to one side, we must do to the other.

Example 1: Basic equation

If \( 2x + 1 = 5 \), what is the value of \( x \)?

\( 2x + 1 = 5 \)

Subtract 1 from both sides: \( 2x = 4 \)

Divide both sides by 2: \( x = 2 \)

Solution: \( x = 2 \)

Example 2: Variables on both sides

If \( 2x - 4 = 5 - x \), what is the value of \( x \)?

\( 2x - 4 = 5 - x \)

Add x to both sides: \( 3x - 4 = 5 \)

Add 4 to both sides: \( 3x = 9 \)

Divide by 3: \( x = 3 \)

Solution: \( x = 3 \)

Fractions and negative numbers

Example 3: Fraction coefficients

If \( \frac{1}{2}x + 3 = 5 \), what is the value of \( x \)?

\( \frac{1}{2}x + 3 = 5 \)

Subtract 3: \( \frac{1}{2}x = 2 \)

Multiply by 2: \( x = 4 \)

Solution: \( x = 4 \)

Example 4: Negative numbers

If \( -2(x - 5) = 1 \), what is the value of \( x \)?

\( -2(x - 5) = 1 \)

Divide by -2: \( x - 5 = -\frac{1}{2} \)

Add 5: \( x = \frac{9}{2} \)

Solution: \( x = 4.5 \)

Try it!

TRY: IDENTIFY THE STEPS TO SOLVING A LINEAR EQUATION

\[7 - 3x = 28\]

To solve the equation above, we can first to both sides of the equation to isolate the x-term.

Next, we can both sides of the equation by -3.

What is the value of \( x \)?

Using linear equations to evaluate expressions

Sometimes you'll need to solve for one expression using information from a linear equation.

Example 5: Evaluating expressions

If \( 2x + 1 = 5 \), what is the value of \( 8x + 4 \)?

Method 1: Solve for x first

\( 2x + 1 = 5 \) → \( x = 2 \)

Then \( 8x + 4 = 8(2) + 4 = 20 \)

Method 2: Notice that \( 8x + 4 = 4(2x + 1) \)

Since \( 2x + 1 = 5 \), then \( 4(2x + 1) = 4(5) = 20 \)

Solution: 20