Linear Inequalities in One or Two Variables

Welcome to our comprehensive lesson on linear inequalities! This fundamental concept appears frequently on the SAT and forms the basis for many problem-solving strategies. Whether you're just starting out or looking to strengthen your skills, understanding how to work with inequalities will significantly boost your mathematical toolkit. In this lesson, we'll explore what linear inequalities are, how to solve them, and how to represent their solutions graphically. By the end, you'll be able to confidently tackle inequality problems on the SAT and beyond.

Linear inequalities are mathematical statements that compare expressions using inequality symbols rather than equals signs. While equations state that two expressions are equal, inequalities indicate that one expression is greater than, less than, greater than or equal to, or less than or equal to another expression.

The four inequality symbols are:

• Less than: $<$
• Greater than: $>$
• Less than or equal to: $\le$
• Greater than or equal to: $\ge$

A linear inequality in one variable has the general form $ax+b<c$ (or using any of the other inequality symbols), where $a$, $b$, and $c$ are constants and $a\ne 0$.

A linear inequality in two variables has the general form $ax+by<c$ (or using any of the other inequality symbols), where $a$, $b$, and $c$ are constants, and at least one of $a$ or $b$ is not zero.

The solution to a linear inequality is not a single value but a range of values that make the inequality true.

Solving Linear Inequalities in One Variable

The process for solving linear inequalities in one variable is similar to solving linear equations, with one important difference: when you multiply or divide both sides by a negative number, you must reverse the inequality symbol.

Here's the step-by-step process:

1. Isolate the variable terms on one side of the inequality.
2. Isolate the constant terms on the other side.
3. Divide both sides by the coefficient of the variable (remember to flip the inequality sign if dividing by a negative number).
4. Express the solution in interval notation or on a number line.

Solving Linear Inequalities in Two Variables

For inequalities in two variables like $ax+by<c$:

1. Rearrange to slope-intercept form: $y<mx+b$ or $y>mx+b$.
2. Graph the boundary line $y=mx+b$ (use a solid line for $\le$ or $\ge$, and a dashed line for $<$ or $>$).
3. Test a point not on the line to determine which side of the line represents the solution.
4. Shade the half-plane that contains the solution.

Systems of Linear Inequalities

To solve a system of linear inequalities:

1. Graph each inequality on the same coordinate plane.
2. Identify the region where all shaded regions overlap.
3. This overlapping region represents the solution to the system.

Part A: Solve the following one-variable inequalities and express the solution in interval notation.

1. $3x-7<8$
2. $2x+5\ge 13$
3. $\frac{x}{4}-3>2$
4. $-2x+6\le -10$
5. $5-3x<11$

Part B: Graph the following two-variable inequalities.

1. $y>2x-3$
2. $y\le -x+4$
3. $3x+2y\ge 6$

Part C: Find the solution region for the following systems of inequalities.

1. $\{\begin{array}{l}x+y\le 5\\ x-y\le 3\\ x\ge 0\\ y\ge 0\end{array}$

2. $\{\begin{array}{l}y>x+1\\ y<-x+5\\ x>0\end{array}$

1. Forgetting to flip the inequality sign: When multiplying or dividing both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol. For example, if $-2x<6$, then $x>-3$ (not $x<-3$).

2. Confusing open and closed intervals: When graphing inequalities with < or >, use an open circle or a dashed line to indicate that the boundary is not included. For ≤ or ≥, use a closed circle or a solid line to show that the boundary is included.

3. Testing the wrong point: When determining which side of a line to shade for a two-variable inequality, make sure to test a point that is clearly not on the line. If you choose a point on the line, you won't be able to determine which side to shade.

4. Assuming all inequalities have solutions: Some systems of inequalities have no solution if the constraints are contradictory. Always check if the solution region exists.

5. Treating inequalities exactly like equations: While many of the same algebraic rules apply, inequalities represent ranges rather than exact values, which requires different thinking about solutions.

6. Misinterpreting "at least" and "at most": "At least" corresponds to ≥, while "at most" corresponds to ≤. Mixing these up leads to incorrect translations of word problems.