Linear Inequality Quiz

1) To solve an inequality means to find all values of the variable for which the statement is true.

True

False

The explanation for the given correct answer is that solving an inequality involves finding all the possible values of the variable that make the statement true. This process often includes using mathematical operations and properties to manipulate the inequality and determine its solution set. By finding these values, one can determine the range of possible solutions for the inequality. Therefore, the statement that solving an inequality means finding all values of the variable for which the statement is true is true.

Explanation

The explanation for the given correct answer is that solving an inequality involves finding all the possible values of the variable that make the statement true. This process often includes using mathematical operations and properties to manipulate the inequality and determine its solution set. By finding these values, one can determine the range of possible solutions for the inequality. Therefore, the statement that solving an inequality means finding all values of the variable for which the statement is true is true.

2) Solve the inequality: 4x + 7 ≥ 2x – 3

X ≥ 2

X ≥ -2

X ≥ -5

X ≤ -5

The correct answer is x ≥ -5 because when we solve the inequality, we subtract 2x from both sides to isolate the variable. This gives us 2x + 7 ≥ -3. Then, we subtract 7 from both sides to get 2x ≥ -10. Finally, we divide both sides by 2 to find that x ≥ -5.

Explanation

The correct answer is x ≥ -5 because when we solve the inequality, we subtract 2x from both sides to isolate the variable. This gives us 2x + 7 ≥ -3. Then, we subtract 7 from both sides to get 2x ≥ -10. Finally, we divide both sides by 2 to find that x ≥ -5.

3) On the graph below, f(x)= -5 (in blue) g(x)= 4-3x (in red) and h(x)= 2 (in green)
f(x) ≤ g(x) ≤ h(x) when 2/3 ≤ x ≤ 3

True

False

The statement "f(x) ≤ g(x) ≤ h(x) when 2/3 ≤ x ≤ 3" means that for all values of x between 2/3 and 3, the value of f(x) is less than or equal to the value of g(x), and the value of g(x) is less than or equal to the value of h(x). Looking at the graph, we can see that the blue line representing f(x) is always below or equal to the red line representing g(x), and the red line is always below or equal to the green line representing h(x) within the given range of x. Therefore, the statement is true.

Explanation

The statement "f(x) ≤ g(x) ≤ h(x) when 2/3 ≤ x ≤ 3" means that for all values of x between 2/3 and 3, the value of f(x) is less than or equal to the value of g(x), and the value of g(x) is less than or equal to the value of h(x). Looking at the graph, we can see that the blue line representing f(x) is always below or equal to the red line representing g(x), and the red line is always below or equal to the green line representing h(x) within the given range of x. Therefore, the statement is true.

4) Solve the following linear inequality graphically: -2x – 6 ≤ 8

X ≤ -7

X ≥ -7

X < 1

X ≥ -1

The correct answer is x ≥ -7. This is because when we graph the inequality -2x - 6 ≤ 8, we can see that the solution lies to the right of -7 on the number line. This means that x is greater than or equal to -7.

Explanation

The correct answer is x ≥ -7. This is because when we graph the inequality -2x - 6 ≤ 8, we can see that the solution lies to the right of -7 on the number line. This means that x is greater than or equal to -7.

5) The following illustration represents the interval [-1, 3)

True

False

The given interval is represented by the notation [-1, 3), which means that it includes all real numbers greater than or equal to -1 and less than 3. The interval includes -1 but does not include 3. Therefore, the statement "The following illustration represents the interval [-1, 3)" is false.

Explanation

The given interval is represented by the notation [-1, 3), which means that it includes all real numbers greater than or equal to -1 and less than 3. The interval includes -1 but does not include 3. Therefore, the statement "The following illustration represents the interval [-1, 3)" is false.

6) Write the following inequalities using interval notation: x < -4 and -2 ≤ x < 7

(-∞, -4) and [-2, 7)

(-4, ∞) and [-2, 7)

(-∞, -4) and (-2, 7)

[-∞, -4] and [-2, 7]

The first inequality, x

Explanation

The first inequality, x

7) Fill in the blank with the correct inequality symbol:

• •If x ≥ 5, then -4x ______ -20

≥

>

≤

=

If x is greater than or equal to 5, then multiplying x by -4 will give a negative number. To find the maximum value for -4x that is less than or equal to -20, we need to find the largest negative number that satisfies this condition. Therefore, the correct inequality symbol to use is ≤.

Explanation

If x is greater than or equal to 5, then multiplying x by -4 will give a negative number. To find the maximum value for -4x that is less than or equal to -20, we need to find the largest negative number that satisfies this condition. Therefore, the correct inequality symbol to use is ≤.

8) A car that averages 25 miles per gallon has a tank that holds 20 gallons of gasoline. After a trip that covered at least 300 miles, the car ran out of gasoline. What is the range of the amount of gasoline (in gallons) that was in the tank at the start of the trip?

15 gallons ≤ x ≤ 20 gallons

12 gallons ≤ x ≤ 20 gallons

15 gallons ≤ x ≤ 25 gallons

0 gallons ≤ x ≤ 12 gallons

12 gallons < x ≤ 20 gallons

The car averages 25 miles per gallon and the trip covered at least 300 miles. Therefore, the car would have needed at least 12 gallons of gasoline (300 miles / 25 miles per gallon) to complete the trip. Since the tank holds 20 gallons, the range of the amount of gasoline in the tank at the start of the trip is 12 gallons ≤ x ≤ 20 gallons.

Explanation

The car averages 25 miles per gallon and the trip covered at least 300 miles. Therefore, the car would have needed at least 12 gallons of gasoline (300 miles / 25 miles per gallon) to complete the trip. Since the tank holds 20 gallons, the range of the amount of gasoline in the tank at the start of the trip is 12 gallons ≤ x ≤ 20 gallons.

9) Two inequalities having exactly the same solution set are called

Two inequalities are called equivalent if they have exactly the same solution set. This means that for any given value, if it satisfies one inequality, it will also satisfy the other inequality. In other words, the two inequalities are interchangeable and represent the same set of values.

Explanation

Two inequalities are called equivalent if they have exactly the same solution set. This means that for any given value, if it satisfies one inequality, it will also satisfy the other inequality. In other words, the two inequalities are interchangeable and represent the same set of values.

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10) A linear inequality in two variables represents a region on a coordinate plane that is bounded by a _______ line.

A linear inequality in two variables, such as "y > 2x + 1," represents a region on a coordinate plane where all the points (x, y) satisfy the inequality. The boundary of this region is a straight line, which is determined by the corresponding linear equation (in this case, "y = 2x + 1"). The line is either solid or dashed, depending on whether the inequality includes the equality sign (≥ or ≤) or not (> or <). The region representing the solutions to the inequality lies on one side of the line, which is determined by testing a point not on the line.

Explanation

A linear inequality in two variables, such as "y > 2x + 1," represents a region on a coordinate plane where all the points (x, y) satisfy the inequality. The boundary of this region is a straight line, which is determined by the corresponding linear equation (in this case, "y = 2x + 1"). The line is either solid or dashed, depending on whether the inequality includes the equality sign (≥ or ≤) or not (> or <). The region representing the solutions to the inequality lies on one side of the line, which is determined by testing a point not on the line.

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