Ask a New Question
Menu

What property of inequality justifies the steps shown in the following solution process?

43d≥−20

34(43)d≥−20(34)

d≥−15

(1 point)
Responses

Subtraction Property of Inequality
Subtraction Property of Inequality

Addition Property of Inequality
Addition Property of Inequality

Division Property of Inequality
Division Property of Inequality

Multiplication Property of Inequality
Multiplication Property of Inequality

What property of inequality can justify the math in Step 2?

Step 1:

−6(3w−7)>−66

−18w+42>−66

Step 2:

−18w+42−42>−66−42

−18w>−108

Step 3:

−18w−18>−108−18

w<6

(1 point)
Responses

Division Property of Inequality
Division Property of Inequality

Subtraction Property of Inequality
Subtraction Property of Inequality

Multiplication Property of Inequality
Multiplication Property of Inequality

Addition Property of Inequality

Juana, Anish, and Giselle are comparing their solutions to the inequality 56x≤−10 . Juana says that the inequality symbol must be reversed because there is a negative sign in the inequality. Anish says the Multiplication Property of Inequality or Division Property of Inequality must be applied. Giselle says the inequality symbol must stay the same.

Based on these answers, which combination of people is correct?

(1 point)
Responses

Juana and Anish
Juana and Anish

Anish, Juana, and Giselle
Anish, Juana, and Giselle

Giselle and Juana
Giselle and Juana

Giselle and Anish
Giselle and Anish

Use the image to answer the question.

A number line ranges from negative 12 to 0 in increments of 1. A closed point is plotted at negative 7. An arrow extends indefinitely left from the closed point past negative 12 along the number line.

What inequality does the number line graph?

(1 point)
Responses

x>−7
x greater than negative 7

x≥−7
x greater-than-or-equal-to negative 7

x<−7
x less than negative 7

x≤−7

Graph x>11 .(1 point)
Responses

A number line ranges from 5 to 15 in increments of 1. An open point is plotted at 11. An arrow extends indefinitely left from the open point past 5 along the number line.
Image with alt text: A number line ranges from 5 to 15 in increments of 1. An open point is plotted at 11. An arrow extends indefinitely left from the open point past 5 along the number line.

A number line ranges from 5 to 15 in increments of 1. A closed point is plotted at 11. An arrow extends indefinitely left from the closed point past 5 along the number line.
Image with alt text: A number line ranges from 5 to 15 in increments of 1. A closed point is plotted at 11. An arrow extends indefinitely left from the closed point past 5 along the number line.

A number line ranges from 5 to 15 in increments of 1. An open point is plotted at 11. An arrow extends indefinitely right from the open point past 15 along the number line.
Image with alt text: A number line ranges from 5 to 15 in increments of 1. An open point is plotted at 11. An arrow extends indefinitely right from the open point past 15 along the number line.

A number line ranges from 5 to 15 in increments of 1. A closed point is plotted at 11. An arrow extends indefinitely right from the closed point past 15 along the number line.

1 answer

Let's go through each question and identify the correct answers:

  1. What property of inequality justifies the steps shown in the following solution process?

    • From: \(43d \geq -20\)
    • To: \(34(43)d \geq -20(34)\)
    • Result: \(d \geq -15\)

    Here, since both sides are multiplied by a positive number (34), we should use the Multiplication Property of Inequality.

    Answer: Multiplication Property of Inequality

  1. What property of inequality can justify the math in Step 2?

    • Step 1: \(-6(3w - 7) > -66\)
    • Simplified: \(-18w + 42 > -66\)
    • Step 2: \(-18w + 42 - 42 > -66 - 42\)
    • Result: \(-18w > -108\)

    Here, in Step 2, we are subtracting 42 from both sides, which justifies using the Subtraction Property of Inequality.

    Answer: Subtraction Property of Inequality

  1. Based on their statements regarding the inequality \(56x \leq -10\), which combination of people is correct?

    • Juana says the inequality must be reversed because there is a negative sign.
    • Anish believes the Multiplication or Division Property must be applied.
    • Giselle says the inequality symbol must stay the same.

    Since both Juana and Anish misinterpret the situation (the sign does not change simply due to being negative on one side), Giselle is actually the one who is correct—the direction of the inequality does not change. Thus, only Giselle is correct.

    Answer: Giselle

  1. What inequality does the number line graph?

    • A closed point at \(-7\) with an arrow extending left indicates values less than or equal to \(-7\).

    Answer: \(x \leq -7\)

  1. Graph \(x > 11\):

    • An open point at 11 with an arrow extending indefinitely to the right indicates that \(x\) can take any value greater than 11.

    Answer: A number line ranges from 5 to 15 in increments of 1. An open point is plotted at 11. An arrow extends indefinitely right from the open point past 15 along the number line.

I hope this helps clarify the solutions! Let me know if you have any further questions.