Master this deck with 19 terms through effective study methods.
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When you multiply or divide both sides of an inequality by a negative number, you must swap the inequality sign. For example, if you have 'x < y' and you multiply both sides by -1, it becomes 'x > y'.
The solution set for x > -120 in interval notation is written as (-120, ∞). This indicates that -120 is not included in the solution set, and all numbers greater than -120 are included.
To solve x/(-3) > -10/9, multiply both sides by -3, which reverses the inequality sign. This results in x < 10/3, so the solution set in interval notation is (-∞, 10/3).
In interval notation, including an endpoint is indicated by a bracket [ ] and excluding it by a parenthesis ( ). For example, [5/3, ∞) includes 5/3, while (10/3, ∞) does not include 10/3.
To solve -0.5x ≤ 7.5, divide both sides by -0.5, which requires swapping the inequality sign. This results in x ≥ -15, meaning the solution set includes all numbers greater than or equal to -15.
To solve 75x ≥ 125, divide both sides by 75, which does not change the inequality sign since 75 is positive. This results in x ≥ 5/3, and the solution set in interval notation is [5/3, ∞).
The notation {x | x > -15} represents the set of all real numbers x such that x is greater than -15. It is a way to describe the solution set using set-builder notation.
To graph the solution set for x < 10/3, draw a number line, mark 10/3, and shade to the left of 10/3, using a parenthesis to indicate that 10/3 is not included in the solution set.
Multiplying both sides of the inequality x/(-15) < 8 by -15 reverses the inequality sign, resulting in x > -120. This indicates that the solution set includes all numbers greater than -120.
The solution set for the inequality x > 8 is expressed in interval notation as (8, ∞). This means all numbers greater than 8 are included in the solution set.
The solution set for x < 10/3 in interval notation is written as (-∞, 10/3). This indicates that all numbers less than 10/3 are included, but 10/3 itself is not.
The coefficient of the variable in an inequality determines how you manipulate the inequality. If the coefficient is positive, you can divide or multiply without changing the inequality sign; if negative, you must swap the sign.
If an inequality has no equal sign, it indicates that the endpoint is not included in the solution set. For example, in x < 5, the value 5 is not part of the solution set.
To determine if a number is part of the solution set for an inequality, substitute the number into the inequality. If the inequality holds true, then the number is part of the solution set.
Dividing both sides of the inequality -0.5x ≤ 7.5 by -0.5 requires swapping the inequality sign, resulting in x ≥ -15. This indicates that all numbers greater than or equal to -15 satisfy the inequality.
The graphical representation of the solution set for x ≥ -15 includes a number line with a closed dot at -15, shading to the right to indicate all numbers greater than or equal to -15 are included.
The term 'solution set' refers to the set of all values that satisfy a given inequality. It can be expressed in various forms, including interval notation, set-builder notation, or graphically.
The solution set for x < -15 in interval notation is expressed as (-∞, -15). This indicates that all numbers less than -15 are included in the solution set.
Multiplying both sides of an inequality by a positive number does not change the direction of the inequality sign. For example, if you have x < 5 and multiply by 2, it remains 2x < 10.