Mastering Linear Inequalities In One Or Two Variables Questions on the SAT

Linear equations and inequalities are fundamental in algebra, involving equal and inequality signs respectively. While linear equations use the equal sign ($=$), linear inequalities use inequality signs ($<$, $\leq$, $>$, and $\geq$).

Linear inequality questions show up on every digital SAT exam. This guide will help you solve linear equations and inequalities, recognize different solution conditions, and ensure you avoid common mistakes.

1. Linear Equations

To understand how to solve linear inequality problems, we have to first understand how to solve their counterpart: linear equations.

The goal of solving a linear equation is to isolate the variable on one side of the equation. This involves treating both sides of the equation equally.

Linear Equations in One Variable

Most SAT questions involve one variable. For example:

$3x + 5 = 14$

Step-by-Step:

1. Subtract 5 from both sides: $3x + 5 - 5 = 14 - 5$ which simplifies to $3x = 9$.
2. Divide by 3: $\frac{3x}{3} = \frac{9}{3}$ which simplifies to $x = 3$.

Combining like terms and distributing coefficients may also be necessary:

$2(x + 3) = 4x - 2$

Step-by-Step:

1. Distribute 2: $2x + 6 = 4x - 2$
2. Subtract 2x from both sides: $2x + 6 - 2x = 4x - 2 - 2x$ which simplifies to $6 = 2x - 2$
3. Add 2 to both sides: $6 + 2 = 2x - 2 + 2$ which simplifies to $8 = 2x$
4. Divide by 2: $\frac{8}{2} = \frac{2x}{2}$ which simplifies to $4 = x$

Fractions and Negative Numbers

Fractions and negative numbers can make equations more complex. When working with fractions, consider clearing them early:

$\frac{2}{3}x = 4$

Step-by-Step:

1. Multiply both sides by 3: $3 \cdot \frac{2}{3}x = 4 \cdot 3$ which simplifies to $2x = 12$.
2. Divide by 2: $\frac{2x}{2} = \frac{12}{2}$ which simplifies to $x = 6$.

With negative numbers, remember to reverse the inequality when multiplying or dividing by a negative.

$-2x + 5 = 9$

Step-by-Step:

1. Subtract 5 from both sides: $-2x + 5 - 5 = 9 - 5$ which simplifies to $-2x = 4$.
2. Divide by -2: $\frac{-2x}{-2} = \frac{4}{-2}$ which simplifies to $x = -2$.

Linear Equations in Two Variables

For equations with two variables, substitute the given value of one variable to find the other:

$3x + 2y = 12$ If $y = 2$, substitute to find $x$.

Step-by-Step:

1. Substitute $y = 2$: $3x + 2(2) = 12$ which simplifies to $3x + 4 = 12$.
2. Subtract 4 from both sides: $3x + 4 - 4 = 12 - 4$ which simplifies to $3x = 8$.
3. Divide by 3: $\frac{3x}{3} = \frac{8}{3}$ which simplifies to $x = \frac{8}{3}$.

Using Linear Equations to Evaluate Expressions

Sometimes, you'll solve an equation and then use that solution to evaluate an expression:

If $2x + 3 = 7$, what is $x - 1?$

Step-by-Step:

1. Solve for $x$: $2x + 3 = 7$.
2. Subtract 3 from both sides: $2x = 4$.
3. Divide by 2: $x = 2$.
4. Evaluate $x - 1$: $2 - 1 = 1$.

Complete Example

Identify the steps to solving a linear equation:

$3x - 5 = 10$

Step-by-Step:

1. Add 5 to both sides: $3x - 5 + 5 = 10 + 5$ which simplifies to $3x = 15$.
2. Divide by 3: $\frac{3x}{3} = \frac{15}{3}$ which simplifies to $x = 5$.

2. Linear Inequalities

Now that we understand how linear equations work, we can see that solving linear inequalities is very similar to solving linear equations. The main difference is that we must pay attention to the direction of the inequality sign.

Linear Inequalities Without Reversing the Inequality Sign

If the coefficient of the variable is positive, the inequality sign maintains its direction:

$2x - 3 < 7$

Step-by-Step:

1. Add 3 to both sides: $2x - 3 + 3 < 7 + 3$ which simplifies to $2x < 10$.
2. Divide by 2: $\frac{2x}{2} < \frac{10}{2}$ which simplifies to $x < 5$.

Linear Inequalities Requiring Reversing the Inequality Sign

If the coefficient is negative, reverse the inequality sign when dividing or multiplying:

$-3x + 4 > 1$

Step-by-Step:

1. Subtract 4 from both sides: $-3x + 4 - 4 > 1 - 4$ which simplifies to $-3x > -3$.
2. Divide by -3 and reverse the sign: $\frac{-3x}{-3} < \frac{-3}{-3}$ which simplifies to $x < 1$.

Complete Example

Identify the steps to solving a linear inequality:

$-4x + 7 \leq 19$

Step-by-Step:

1. Subtract 7 from both sides: $-4x + 7 - 7 \leq 19 - 7$ which simplifies to $-4x \leq 12$.
2. Divide by -4 and reverse the sign: $\frac{-4x}{-4} \geq \frac{12}{-4}$ which simplifies to $x \geq -3$.

3. Number of Solutions for Linear Equations

Linear equations on the SAT typically have one solution, but some can have no solutions or infinitely many solutions.

One Solution

If the equation can be rewritten in the form $ax = b$, where $a$ and $b$ are constants, it has one solution:

$3x + 2 = 8$

Step-by-Step:

1. Subtract 2 from both sides: $3x + 2 - 2 = 8 - 2$ which simplifies to $3x = 6$.
2. Divide by 3: $\frac{3x}{3} = \frac{6}{3}$ which simplifies to $x = 2$.

No Solutions

If the variable can be eliminated, leaving a false statement, there are no solutions:

$2x + 3 = 2x + 5$

Step-by-Step:

1. Subtract 2x from both sides: $2x + 3 - 2x = 2x + 5 - 2x$ which simplifies to $3 = 5$ which is false.

Infinitely Many Solutions

If the equation simplifies to a true statement involving no variables, there are infinitely many solutions:

$x + 3 = x + 3$

Step-by-Step:

1. Subtract $x$ from both sides: $x + 3 - x = x + 3 - x$ which simplifies to $3 = 3$ which is always true.

Complete Example

Determine the number of solutions for the following equation:

$4x + 2 = 4(x + 1)$

Step-by-Step:

1. Distribute 4: $4x + 2 = 4x + 4$
2. Subtract 4x from both sides: $4x + 2 - 4x = 4x + 4 - 4x$ which simplifies to $2 = 4$ which is false.

The equation has no solutions.