4.2 Solving Linear Equations and Inequalities

Solving Linear Equations

A linear equation is a mathematical statement asserting that two algebraic expressions are equal. Solving a linear equation means finding the value of the variable that makes the statement true. This value is called the "solution" or the "root" of the equation. On the SHSAT, you must be able to solve both simple one-step equations and complex multi-step equations quickly and accurately.

The Balancing Rule of Algebra

The fundamental rule of solving equations is that the equation must always remain balanced. Whatever mathematical operation you perform on one side of the equation, you must perform the exact same operation on the other side. Think of a balance scale: if you add weight to one side, you must add the same weight to the other to keep it level.

To isolate the variable (get it by itself on one side of the equal sign), you use "inverse operations." Inverse operations are operations that undo each other:

• Addition and Subtraction are inverses.
• Multiplication and Division are inverses.

For example, to solve $x + 5 = 12$, you undo the addition of $5$ by subtracting $5$ from both sides, yielding $x = 7$. To solve $3x = 15$, you undo the multiplication by $3$ by dividing both sides by $3$, yielding $x = 5$.

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Solving Multi-Step Equations

Multi-step equations require multiple operations to isolate the variable. The general strategy is:

1. Clear fractions or decimals by multiplying all terms by a common denominator or power of 10.
2. Simplify each side by distributing and combining like terms.
3. Move variable terms to one side of the equation (usually the left) and constant terms to the other.
4. Isolate the variable using division or multiplication.

Clearing Fractions: A High-Yield Strategy

When an equation contains multiple fractions, working with common denominators can be slow and prone to errors. You can clear the fractions entirely by multiplying the entire equation by the Least Common Multiple (LCM) of all the denominators.

• Example: Solve for $x$: $\frac{2}{3}x - 5 = \frac{1}{4}x + 2$ The denominators are $3$ and $4$. The LCM of $3$ and $4$ is $12$. Multiply every term on both sides by $12$: $12 \left(\frac{2}{3}x\right) - 12(5) = 12 \left(\frac{1}{4}x\right) + 12(2)$ $8x - 60 = 3x + 24$ Now, move the variable terms to the left by subtracting $3x$ from both sides: $5x - 60 = 24$ Next, move the constants to the right by adding $60$ to both sides: $5x = 84$ Finally, divide both sides by $5$: $x = \frac{84}{5} \text{ or } 16.8$ Clearing fractions makes the algebra much simpler and faster.

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Solving Linear Inequalities

An inequality is a mathematical statement that compares two expressions using inequality symbols:

• $<$ (less than)
• $\le$ (less than or equal to)
• $>$ (greater than)
• $\ge$ (greater than or equal to)

Solving an inequality is almost identical to solving an equation, with one crucial exception: The Negative Rule of Inequalities.

The Negative Rule

[!IMPORTANT] When you multiply or divide both sides of an inequality by a negative number, you must reverse (flip) the direction of the inequality sign.

To understand why this is true, look at a simple number line comparison: $2 < 5$ If we multiply both sides by $-1$, we get $-2$ and $-5$. On a number line, $-2$ is to the right of $-5$, which means $-2$ is greater than $-5$: $-2 > -5$ The inequality sign had to flip to keep the statement true.

• Example: Solve $7 - 3x \le 22$. First, subtract $7$ from both sides: $-3x \le 15$ Now, divide both sides by $-3$. Because you are dividing by a negative number, you must flip the inequality sign from $\le$ to $\ge$: $x \ge -5$ If you forget to flip the sign, your solution set will be entirely incorrect.

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Graphing Solution Sets on a Number Line

Since inequalities have infinitely many solutions, we visualize them by graphing the solution set on a number line.

Key Graphing Rules

• Circle Type:
  • Use an open circle ($\circ$) for strict inequalities ($<$ or $>$). This indicates that the endpoint is not included in the solution set.
  • Use a closed (solid) circle ($\bullet$) for non-strict inequalities ($\le$ or $\ge$). This indicates that the endpoint is included.
• Shading Direction:
  • Shade to the right for greater than ($>$ or $\ge$).
  • Shade to the left for less than ($<$ or $\le$).

Compound Inequalities

A compound inequality is two inequalities joined together. On the SHSAT, you will frequently see "AND" compound inequalities, which represent a range of values.

• Example: Solve $-3 < 2x + 1 \le 9$. To solve, perform operations on all three parts of the inequality simultaneously. Subtract $1$ from all three parts: $-4 < 2x \le 8$ Divide all three parts by $2$: $-2 < x \le 4$ This means $x$ is greater than $-2$ AND less than or equal to $4$.
• Graphing this solution: Place an open circle at $-2$ (not included) and a closed circle at $4$ (included). Shade the line between these two endpoints to represent all real numbers that satisfy both conditions.