Solving Linear Equations
Solving linear equations involves isolating the variable and using inverse operations to rearrange the terms of the equation. This process ensures that the variable is alone on one side of the equation, allowing you to solve for its value. For example, try to solve for $x$ in the following equation:
$3x + 5 = 20$
Subtract 5 from both sides to eliminate the constant term:
$3x + 5 − 5 = 20 − 5 → 3x = 15$
Divide both sides by 3 to isolate $x$:
$\dfrac{3x}{3} = \dfrac{15}{3} → x = 5$
This step-by-step approach helps to clearly identify and remove terms that obstruct the isolation of the variable.
In more complex equations, there may be additional steps required to simplify terms before solving. For example, try solving for $x$ in this equation:
$4(x − 2) = 3x + 6$
Distribute 4 to eliminate the parentheses:
$4x − 8 = 3x + 6$
Subtract 3$x$ from both sides to group all terms with $x$ on one side:
$4x − 3x − 8 = 3x − 3x + 6 $ $ → x − 8 = 6$
Add 8 to both sides to isolate $x$:
$x − 8 + 8 = 6 + 8 → x = 14$
By following these steps, the solution is found systematically and accurately.
Solving Inequalities
Solving inequalities follows similar steps as solving equations but includes an important rule: If you multiply or divide by a negative number, you must reverse the inequality sign. For example, consider the following inequality:
$−2x > 4$
Divide both sides by −2 to isolate x, remembering to reverse the inequality:
$x < −2$
This rule ensures the inequality remains accurate after applying an operation that changes the direction of the values.
Additionally, inequalities can have solutions represented as ranges. Let’s look at an example:
Solve $2x − 3 ≤ 5$
Add 3 to both sides:
$2x ≤ 8$
Divide both sides by 2:
$x ≤ 4$
The solution is $x ≤ 4$, which can also be expressed graphically on a number line or using interval notation $(−∞, 4]$. These additional methods help students visualize solutions and apply them in context.
Key Tip: When solving equations or inequalities, always check your solution by substituting your answer back into the original problem. This ensures your solution satisfies the equation or inequality and reduces the likelihood of errors.