B.E.S.T. Algebra 1 EOC: a complete guide to linear and absolute-value equations, inequalities, and systems

What this category demands

This guide covers the Algebra and Modeling reporting category, about 41 percent of the B.E.S.T. Algebra 1 EOC and its single largest block. The skills are solving linear equations and inequalities, writing and graphing lines, absolute-value equations and inequalities, and systems. These points are the most reliable on the test, so securing them is the surest path to Level 3. Each dot-point page has its own practice: solving linear equations, solving linear inequalities, writing and graphing linear functions, absolute-value equations and inequalities, systems of linear equations, and systems of linear inequalities.

Solving linear equations

Keep the equation balanced, doing the same operation to both sides, and undo operations in reverse order. Clear fractions by multiplying every term by the common denominator, and gather variables on both sides onto one side first. When the variable cancels, a true leftover ($5 = 5$) means infinitely many solutions and a false leftover ($8 = 5$) means no solution, the algebraic shadow of identical versus parallel lines.

Solving linear inequalities

Solve exactly like equations, except flip the inequality sign when you multiply or divide by a negative. Graph with an open circle for strict $<$ or $>$ and a closed circle for $\le$ or $\ge$, shading toward the solutions. In context, a budget or count usually restricts the answer to whole numbers.

Writing and graphing linear functions

Slope is $m = \frac{y_2 - y_1}{x_2 - x_1}$. Use slope-intercept $y = mx + b$ to read features, point-slope $y - y_1 = m(x - x_1)$ to build from a slope and a point, and standard form $Ax + By = C$ when needed. Parallel lines share a slope; perpendicular lines have opposite-reciprocal slopes (product $-1$).

Absolute-value equations and inequalities

Isolate the bars, then split into two cases ($= k$ or $= -k$). Inequalities: $|x| < k$ is the between "and" compound $-k < x < k$; $|x| > k$ is the outside "or" compound $x < -k$ or $x > k$. An absolute value equal to a negative has no solution.

Systems of equations and inequalities

A system of equations is solved by the point satisfying both, by graphing, substitution, or elimination; outcomes are one solution (crossing), none (parallel), or infinitely many (same line). A system of inequalities is graphed as overlapping half-planes (solid boundary for $\le, \ge$; dashed for $<, >$), and the solution is the overlap region; a point is a solution only if it satisfies every inequality.

How this category is examined

• Equation editor. Solve equations, inequalities, and systems, typing values, inequalities, or ordered pairs.
• GRID and hot-spot. Graph lines, number-line solutions, and inequality regions; place circles and shading.
• Multiple choice and multiselect. Count solutions, identify slopes, test points, or pick the feasible region.
• Context items. Model budgets, tickets, and constraints, then solve and interpret.

Check your knowledge

Work these as you would for credit on the computer-based test.

1. Solve $6x - 5 = 2x + 19$. (2 points)
2. How many solutions does $2(x + 4) = 2x + 8$ have? (1 point)
3. Solve $-4x + 3 \ge 19$. (2 points)
4. Write the slope-intercept equation through $(0, -3)$ with slope $4$. (1 point)
5. What slope is perpendicular to $y = \frac{1}{2}x + 6$? (1 point)
6. Solve $|x - 5| = 8$. (2 points)
7. Write $|x| \ge 4$ as a compound inequality. (1 point)
8. Solve the system $y = 3x$ and $x + y = 8$. (2 points)
9. Is $(0, 0)$ a solution of $y < 2x - 1$? (1 point)