STAAR Algebra I: a complete guide to writing and solving linear functions, equations, and inequalities

What this category demands

The Writing and Solving Linear Functions, Equations, and Inequalities reporting category (TEKS A.2, A.5, A.12) is the largest on the STAAR Algebra I test, about 29 percent of the points. It rewards fluent solving and clear modeling: solve equations and inequalities, find domain and range, build a linear function from a context, evaluate function notation, and solve and model with systems. This guide ties together the dot-point pages, each with its own practice: solving linear equations, solving linear inequalities, domain and range of linear functions, writing and modeling with linear functions, solving systems of linear equations, and modeling with systems of equations.

Solving equations and inequalities

Solve a linear equation by undoing operations in reverse: distribute, combine like terms, collect the variable, divide. Inequalities use the same steps with one rule: flip the sign when multiplying or dividing by a negative. When the variable cancels, a true statement means infinitely many solutions and a false statement means none. Graph an inequality's solution on a number line with an open circle for strict and closed for inclusive, the ray pointing toward the solutions.

Domain, range, and function notation

The domain is the set of inputs, the range the set of outputs. A bare non-vertical line has domain and range of all real numbers; a real-world situation restricts to reasonable values, continuous (intervals, for time or distance) or discrete (whole numbers, for counts), written with inequalities. Function notation $f(x)$ names the output for an input: $f(3)$ is found by substituting $x = 3$.

Writing linear models

A linear model is $f(x) = (\text{rate})x + (\text{initial value})$: the per-unit "rate" is the slope, the one-time or starting amount is the $y$-intercept. Read a description for the recurring quantity (per month, per mile) and the fixed quantity (a fee, a starting balance). Build the function, then either evaluate (input given) or solve $f(x) = k$ (output given).

Systems of equations

A system asks for the point satisfying both equations. Solve by graphing (read the intersection), substitution (a variable is isolated), or elimination (add or subtract to cancel, scaling first if needed). The number of solutions matches the lines: one (crossing), none (parallel), or infinitely many (same line). To model a system, define variables and write one equation per condition, often a count equation and a value equation; a break-even problem sets cost equal to revenue.

How this category is examined

• Multiple choice. Solve equations, inequalities, and systems; choose a model. The "forgot to flip", "distribute to first term", and rate-versus-fee distractors are standard.
• Equation editor and number entry. Build a function or system, enter a solution value or ordered pair, or a break-even quantity.
• Hot spot. Graph an inequality on a number line (circle style and ray direction).
• Inline choice. Classify the number of solutions, continuous versus discrete, or interpret a slope.

Check your knowledge

Work these as you would for credit on the redesigned test.

1. Solve $4(x - 3) = 2x + 6$. (1 point)
2. How many solutions does $2(x + 4) = 2x + 8$ have? (1 point)
3. Solve $-2x + 7 > 1$. (1 point)
4. State the domain (continuous or discrete) for buying up to 15 tickets. (1 point)
5. A taxi charges $5 plus$3 per mile. Write $C(m)$ and find $C(4)$. (2 points)
6. Solve the system $y = 2x + 1$ and $3x + y = 16$. (2 points)
7. How many solutions do $y = 3x - 2$ and $y = 3x + 4$ have? (1 point)
8. Adult tickets $10, child$6; 9 tickets total cost $70. Write the system. (2 points)