To solve 4x - 9 ≤ 3x + 2, we need to isolate x. Subtracting 3x from both sides gives x - 9 ≤ 2. Adding 9 to both sides yields x ≤ 11. The key technique is to move all x terms to one side and constants to the other. A common error is sign mistakes when moving terms - remember that subtracting 3x from both sides is different from adding -3x. Always verify by substituting: if x = 11, then 4(11) - 9 = 35 and 3(11) + 2 = 35, so 35 ≤ 35 ✓.

A closed circle at 1 and open circle at 5, with shading to the left of 1 and right of 5, represents values less than or equal to 1 OR greater than 5. The closed circle at 1 means 1 is included (≤), while the open circle at 5 means 5 is not included (>). Since the shading is in two separate regions, this is an OR compound inequality: x ≤ 1 or x > 5. The key indicator of OR is disconnected shading regions. When you see shading in multiple separate areas, use OR to connect the inequalities.

Open circles at both -2 and 4 with shading between them represents all values strictly between -2 and 4. The open circles mean that neither -2 nor 4 is included in the solution set, giving us the strict inequalities. Since the shading is between the points (not outside), this is an AND situation, not OR. The compound inequality is -2 < x < 4. A common error is confusing shading between points (AND) with shading outside points (OR). When shading is continuous between two points, it's always an AND compound inequality.

An open circle at 3 with shading to the left represents all values less than 3, but not including 3 itself. The open circle indicates that 3 is NOT part of the solution set, which corresponds to the strict inequality <. Shading to the left means values less than 3. Therefore, the inequality is x < 3. The key distinction is open circle (< or >) versus closed circle (≤ or ≥). When matching graphs to inequalities, always check both the circle type and shading direction.

A closed dot at -1 with shading to the right represents all values greater than or equal to -1. The closed dot indicates that -1 is included in the solution set, which corresponds to the "or equal to" part of ≥. Shading to the right means values greater than -1. Therefore, the inequality is x ≥ -1. Remember: closed dot means ≤ or ≥ (inclusive), while open dot means < or > (exclusive). The direction of shading indicates whether values are greater than or less than the boundary point.

The total cost is $40 + $0.10x, and this must be less than $55, giving us 40 + 0.10x < 55. Subtracting 40 from both sides: 0.10x < 15. Dividing by 0.10 (or multiplying by 10): x < 150. Since x must be a whole number of texts, the greatest value is x = 149. The key is recognizing that "less than" (not "less than or equal to") means we cannot include 150. When finding the "greatest whole number" with a strict inequality, you must round down.

To solve 7 - 2x > 15, we first subtract 7 from both sides to get -2x > 8. Now, when dividing both sides by -2, we MUST reverse the inequality sign, giving us x < -4. This is a critical rule: whenever you multiply or divide an inequality by a negative number, the inequality symbol flips. A common error is forgetting to flip the sign, which would incorrectly give x > -4. Always double-check by testing a value: if x = -5, then 7 - 2(-5) = 7 + 10 = 17 > 15 ✓.

The student needs an average of at least 80 on 5 tests. The average is calculated by adding all scores and dividing by the number of tests, so we need (76 + 84 + 79 + 88 + x)/5 ≥ 80. This ensures the mean of all five scores meets the requirement. A common error is dividing by 4 instead of 5 (forgetting to count the unknown score) or not dividing at all. Remember that "average" always means sum divided by count.

The truck can carry at most 1,200 lb, and the loaded weight is 3,800 + 50x. Setting up the inequality: 3,800 + 50x ≤ 5,000 (since 3,800 + 1,200 = 5,000 total capacity). Solving: 50x ≤ 1,200, so x ≤ 24. Since x must be a whole number of boxes, the greatest integer value is x = 24. Always check: 3,800 + 50(24) = 3,800 + 1,200 = 5,000 lb, which equals the maximum capacity. When asked for the "greatest integer value," round down from your decimal answer.

We need to solve both inequalities and find where they overlap. For 2x + 1 < 9: subtract 1 to get 2x < 8, then divide by 2 to get x < 4. For x - 4 ≥ -1: add 4 to get x ≥ 3. Since we need BOTH conditions true (AND), we need the overlap: x must be at least 3 AND less than 4, which gives us 3 ≤ x < 4. The common error is treating AND as OR - with AND, we need the intersection of solution sets, not the union.