A7a
Practice - Algebraic Inequalities
TRY TO ANSWER IN TWO AND A HALF MINUTES.
QUESTIONS
QUESTIONS
1. Solve the inequality -12x + 24.6 ≤ 11.98 x + 14¾.
2. If 12(x - 2) ≤ 24, solve for x?
3. The perimeter of a square equals four times the length of a side. The perimeter of a particular square is less than or equal to 64 inches. Write an inequality describing the length of a side of the square.
4. A compound inequality combines two different inequalities. For example, 3x + 4 < 5x < 16 + x. Simplify this.
5. Solve -4 (x + 2) < 24.
6. You want to save $1800. You work for three months, 20 hours a week, and are paid $9 per hour. Is what you make (less than, equal to, or greater than) your savings goal?
7. A ramp must replace two steps, each of which is 7" high. The slope of the ramp has to have 12" of length for every 1" of height. Is the length of the ramp (greater than, equal to, or less than) 14 feet?
8. Simplify the inequality (3x + 9)/2 < 9.
ANSWERS
1. First convert each term to a common decimal form: -12.00x + 24.60 ≤ 11.98x + 14.75. Then - 0.02x ≤ - 9.85. Multiplying by -1 changes the sign of the inequality, so +0.02x ≥ +9.85. The answer is x ≥ 9.85/0.02 , or 492.5.
2. Multiplying through the parentheses gives 12x - 24 ≤ 24. Adding 24 to both sides of the inequality does not change the sign, so 12x ≤ 48, or x ≤ 12.
3. P = 4s, where P is the perimeter and s is the side. So P ≤ 64, 4s ≤ 64, and s ≤ 16.
4. Separate the two inequalities. The first one says 3x + 4 < 5x. So 4 < 2x and 2 < x. Then look at the second one: 5x < 16 + x, so 4x < 16, and x < 4. Combining the two simplifications gives 2 < x < 4.
5. -4 (x + 2) < 24, so -4x - 8 < 24, and -4x < 32, so -x < 8. Next you have to multiply by -1, which changes the sign of the inequality. So x > -8. Check the answer by plugging -8 back into the inequality.
6. There are 13 weeks (not 12) in three months. So 13 weeks x 20 hours/week x $9/hour = $2340. Your earnings are greater than the $1800 goal.
7. The ramp must be 14" high to replace the steps. Since each inch of height requires 12" of length, the length must be 14 x 12", which is 14 feet. Therefore the answer is "equal to" 14 feet.
8. Just as with an equation, you simplify an inequality by doing the same thing to both sides. The inequality simplifies to (3x + 9 < 9(2) = 18, so 3x < 9 and x < 3. You do not change the direction of the inequality because you did not multiply or divide by a negative number.
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