Virginia SOL Algebra I: a complete guide to equations and inequalities (A.EI)

What this category demands

This guide covers the Equations and Inequalities reporting category, the A.EI standards, about 18 of the 50 operational items and the single largest block on the Virginia Algebra I SOL. The skills are solving linear equations, rearranging literal equations, solving inequalities, absolute value, and systems (quadratic equations, also A.EI, have their own module). These points are the most reliable on the test, so securing them is the surest path to passing. Each dot-point page has its own practice: solving linear equations, literal equations and formulas, solving linear inequalities, 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.

Literal equations and formulas

To solve a formula for a variable, treat the other letters as constants and isolate the target with inverse operations, exactly as you solve a numerical equation. Clear fractions, move added terms, then divide off the coefficient. If the variable appears twice, collect those terms and factor it out before dividing. The answer is an expression, not a number.

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.

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

• Fill-in-the-blank. Solve equations, inequalities, systems, and literal equations, typing values, inequalities, ordered pairs, or rearranged formulas.
• Coordinate-plane and hot-spot. Graph lines, number-line solutions, and inequality regions; place circles and shading.
• Multiple choice. Count solutions, identify boundary styles, test points, or pick the solution region.
• Context items. Model budgets, tickets, and constraints, then solve and interpret.

Check your knowledge

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

1. Solve $6x - 5 = 2x + 19$. (2 points)
2. How many solutions does $2(x + 4) = 2x + 8$ have? (1 point)
3. Solve $V = lwh$ for $h$. (1 point)
4. Solve $-4x + 3 \ge 19$. (2 points)
5. Solve $|x - 5| = 8$. (2 points)
6. Write $|x| \ge 4$ as a compound inequality. (1 point)
7. Solve the system $y = 3x$ and $x + y = 8$. (2 points)
8. Two lines have the same slope but different y-intercepts. How many solutions? (1 point)
9. Should the boundary of $y < 2x - 1$ be solid or dashed? (1 point)
10. Is $(0, 0)$ a solution of $y \le x + 2$? (1 point)