9.1 Solving and Graphing Linear Inequalities in One Variable

Why Linear Inequalities Matter on the GED

Assessment target A.3 ("Write, manipulate, solve, and graph linear inequalities") lives inside Algebraic Problem Solving, which is worth 55% of your GED Mathematical Reasoning score — the single largest content area on the test. A.3 breaks into four official indicators: solving one-variable inequalities (A.3.a), graphing solutions on a number line (A.3.b), solving real-world inequality problems (A.3.c), and writing inequalities from context (A.3.d). This section covers A.3.a and A.3.b — the mechanical skills of solving and graphing. The next section (9.2) covers translating word problems into inequalities.

An inequality is a mathematical statement that compares two expressions using one of four symbols instead of an equals sign. Unlike an equation, which usually has exactly one solution, an inequality has a solution set — a whole range of values that make the statement true.

The Four Inequality Symbols

The GED's official Assessment Guide limits A.3 to these four symbols — the "not equal to" symbol (≠) is not part of this assessment target, so you will not see it in inequality-solving items.

Solving Linear Inequalities: Same Rules, One Critical Exception

Solving a one-variable linear inequality uses the exact same steps as solving a linear equation: distribute, combine like terms, and isolate the variable by adding, subtracting, multiplying, or dividing both sides. The one rule that has no equation equivalent is this:

When you multiply or divide both sides of an inequality by a negative number, you must flip (reverse) the inequality symbol.

Worked Example 1 (no sign flip needed): Solve 3x − 7 < 11.

• Add 7 to both sides: 3x < 18
• Divide both sides by 3 (positive, so the symbol stays the same): x < 6

Worked Example 2 (sign flip required): Solve −2x + 5 ≥ −9.

• Subtract 5 from both sides: −2x ≥ −14
• Divide both sides by −2. Because you divided by a negative number, flip ≥ to ≤: x ≤ 7

If you forget to flip the sign in Example 2, you would (incorrectly) get x ≥ 7 — the exact opposite solution set. This single error is the most common mistake test-takers make on GED inequality items, because the flipping step is easy to do automatically for equations (where it doesn't matter) and easy to forget for inequalities (where it matters completely).

Worked Example 3 (distribution first): Solve −4(x − 2) < 20.

• Distribute: −4x + 8 < 20
• Subtract 8: −4x < 12
• Divide by −4 and flip the symbol: x > −3

Worked Example 4 (rational number coefficients): Because A.3.a specifically covers "rational number coefficients," expect inequalities with fractions or decimals, not just whole numbers. Solve (2/3)x + 1 ≤ 7.

• Subtract 1 from both sides: (2/3)x ≤ 6
• Multiply both sides by the reciprocal, 3/2 (a positive number, so no flip): x ≤ 9

If a fraction coefficient were negative — for example, solving −(3/4)x > 9 — you would still multiply both sides by the reciprocal (−4/3), and because that multiplier is negative, the symbol flips: x < −12.

Graphing Solutions on a Number Line

Indicator A.3.b asks you to identify or draw the graph of an inequality's solution set on a number line, and this often appears as a hot spot item (click the correct point/shading) or a drag-and-drop item (drag a circle and an arrow into place) rather than plain multiple choice.

To graph x > −3:

1. Locate −3 on the number line.
2. Draw an open circle at −3 (because > does not include −3 itself).
3. Shade or draw an arrow to the right (toward larger numbers) since the solution is everything greater than −3.

To graph x ≤ 7:

1. Locate 7 on the number line.
2. Draw a closed (filled) circle at 7 (because ≤ includes 7).
3. Shade or draw an arrow to the left (toward smaller numbers).

A quick way to check your shading direction: pick a test point from your shaded region and plug it into the original inequality. For x > −3, test x = 0: is 0 > −3? Yes — so shading to the right of −3 is correct.

Common Traps to Avoid

• Forgetting to flip the sign after multiplying or dividing by a negative number (the #1 error on this topic).
• Mixing up open and closed circles — an open circle at the boundary means that exact value is not a solution; a closed circle means it is.
• Shading the wrong direction — always re-read the final inequality (e.g., x > −3 shades right, x < −3 shades left) rather than guessing from the original problem's direction.
• Flipping the sign when you shouldn't — only flip when multiplying/dividing by a negative. Adding or subtracting any number, positive or negative, never triggers a flip.

Key Takeaways

Solving inequalities is nearly identical to solving equations, with one make-or-break rule: reverse the inequality symbol only when multiplying or dividing by a negative number. Pair every solved inequality with a number-line check — open circle for strict inequalities, closed circle for "or equal to" — and verify your shading direction with a test point before moving on.