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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

Multiplication Property of Inequality
Multiplication Property of Inequality

Division Property of Inequality
Division Property of Inequality

Addition Property of Inequality

7 answers

The property of inequality that justifies the steps shown in the solution process is the Multiplication Property of Inequality.

In the solution, the original inequality \(43d \geq -20\) is multiplied by 34, which is a positive number. According to the multiplication property, multiplying both sides of an inequality by a positive number does not change the direction of the inequality, which is why the inequality still holds true after multiplying.

So the answer is: 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

Multiplication Property of Inequality
Multiplication Property of Inequality

Subtraction Property of Inequality
Subtraction Property of Inequality

Addition Property of Inequality
Addition Property of Inequality

Division Property of Inequality

The property of inequality that justifies the math in Step 2 is the Subtraction Property of Inequality.

In Step 2, 42 is subtracted from both sides of the inequality \(−18w + 42 > -66\). The Subtraction Property of Inequality states that if you subtract the same number from both sides of an inequality, the inequality remains valid.

So, the answer is: Subtraction 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 Anish
Giselle and Anish

Giselle and Juana

To solve the inequality \(56x \leq -10\), we need to divide both sides by \(56\).

The correct understanding of the properties is as follows:

  • Juana is incorrect. The presence of a negative sign does not automatically mean the inequality symbol must be reversed. The symbol is only reversed when you multiply or divide both sides of an inequality by a negative number.
  • Anish is partially correct because applying the Division Property of Inequality is necessary to isolate \(x\). However, he does not mention the crucial detail about not reversing the inequality when dividing by a positive number (56 in this case).
  • Giselle is correct as well. The inequality symbol will stay the same when dividing by a positive number.

Since Anish is correct about applying the Division Property, and Giselle is correct about the inequality symbol staying the same when dividing by a positive, the correct combination is:

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-or-equal-to negative 7

x<−7
x less than negative 7

x≤−7
x less-than-or-equal-to negative 7

x>−7

The number line you described indicates a closed point plotted at \(-7\) and an arrow extending indefinitely to the left. This implies that all values less than or equal to \(-7\) are included in the solution.

Since the closed point shows that \(-7\) is included in the solution, the correct inequality that represents this is:

\(x \leq -7\) (x less-than-or-equal-to negative 7).