7.1 Expressions, Equations, and Inequalities

GED Focus: Algebra Is the Biggest Slice

Algebraic reasoning is the single largest content group on the GED Mathematical Reasoning test, accounting for about 55 percent of the 46 questions. You have 115 minutes total, and the on-screen TI-30XS MultiView calculator is locked for the first 5 questions, so you must be able to simplify and solve simple linear equations by hand. The passing score is 145 on the 100-200 scale (about 60 to 65 percent of available points).

The official assessment targets here are: write, evaluate, and compute with expressions; solve linear equations and inequalities with rational coefficients; and graph one-variable inequalities on a number line.

Because algebra is so heavily weighted, building speed and accuracy here is the highest-leverage way to clear 145. Treat every algebra item as a chance to bank points you may need to offset harder geometry or data questions later in the test.

Expressions Are Instructions, Not Problems to Solve

An expression has no equals sign. You can simplify or evaluate it, but you cannot solve it. The most common GED error is trying to set an expression equal to zero out of habit. Treat 3(2x - 5) + 4x as a set of instructions: distribute first to get 6x - 15 + 4x, then combine like terms to get 10x - 15.

When you evaluate, substitute the value in parentheses before doing arithmetic so you respect signs: evaluating -2x^2 when x = 3 gives -2(3)^2 = -2(9) = -18, not (-6)^2.

Worked Example: Solve a Linear Equation

Solve 4(x - 3) + 7 = 2x + 11.

1. Distribute the 4 across both terms: 4x - 12 + 7 = 2x + 11.
2. Combine like terms on the left: 4x - 5 = 2x + 11.
3. Subtract 2x from both sides: 2x - 5 = 11.
4. Add 5 to both sides: 2x = 16.
5. Divide both sides by 2: x = 8.

Check by substitution, the move that saves the most points on test day: 4(8 - 3) + 7 = 4(5) + 7 = 27, and 2(8) + 11 = 27. Both sides match, so x = 8 is confirmed. If the equation contained fractions such as (x/3) + 2 = 5, you would clear the denominator by multiplying every term by 3 first.

Inequalities Need Direction

An inequality compares values instead of forcing them equal. Solve it exactly like an equation, with one rule: the symbol reverses when you multiply or divide both sides by a negative number. Adding or subtracting a negative does NOT flip it.

Solve -3x + 5 < 20.

• Subtract 5 from both sides: -3x < 15.
• Divide both sides by -3 and flip the symbol: x > -5.

Graph it with an open circle at -5 (because -5 is not included) and shade to the right. Use a closed circle only when the symbol is >= or <=.

Watch the phrase-to-symbol map, a frequent GED trap:

• "At least" means >= (greater than or equal to).
• "At most" or "no more than" means <= (less than or equal to).
• "More than" or "exceeds" means > (strict).
• "Fewer than" or "less than" means < (strict).

Test-Day Strategy and Common Traps

Read the wording before touching the calculator; words such as "the sum of" (add), "the product of" (multiply), and "the difference of" (subtract in order) set the structure. Remember that "5 less than x" is x - 5, never 5 - x, because the order reverses.

Most missed GED algebra items come from three habits:

1. Distributing to only the first term inside parentheses.
2. Dropping a negative sign while moving terms across the equals sign.
3. Forgetting to flip the inequality when dividing by a negative.

Keep your scratch-paper work vertical, line up the equals or inequality signs in a column, and on multiple-choice items plug the most reasonable answer choice back into the ORIGINAL sentence before committing. The format may be drag-and-drop, fill-in-the-blank, or multiple choice, but the underlying algebra rules never change.

Order of Operations When Solving

Solving an equation reverses the order of operations. To isolate x, undo addition and subtraction first, then undo multiplication and division last. In 5x + 3 = 23, subtract 3 before dividing by 5: 5x = 20, then x = 4. Reversing that order, dividing by 5 while the 3 is still attached, is a common error. For two-sided equations such as 7x - 2 = 3x + 14, first collect variable terms on one side (subtract 3x to get 4x - 2 = 14), then constants, then divide. Keeping this discipline matters most on the first 5 calculator-free questions, where a clean by-hand process prevents arithmetic slips you cannot lean on the calculator to catch.