The real number satisfies both of the following inequalities: and What is the greatest integer value of that satisfies both inequalities?

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SAT Linear Inequalities Question 111 | Free SAT Prep | Aniko

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Question 111·Hard·Linear Inequalities in One or Two Variables

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The real number $x$ satisfies both of the following inequalities:

3(2 - x) + 8 < 5x - 4

and

2x - 5 \ge 3(x - 4).

What is the greatest integer value of $x$ that satisfies both inequalities?

Your answer

(Express the answer as an integer)

For linear inequalities, always solve each inequality separately by first distributing, then combining like terms, and then isolating $x$ . Watch the direction of the inequality sign, especially if you divide or multiply by a negative (you must flip the sign). Once you have each solution written simply (like $x > a$ or $x \le b$ ), interpret “satisfies both” as the intersection (overlap) of the solution sets. Only at the end, when the question asks for an integer, list or reason through the integers in that overlap and pick the one that fits the question (here, the greatest integer).

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Work with each inequality separately

Treat each inequality one at a time. First, fully simplify and solve $3(2 - x) + 8 < 5x - 4$ , then separately simplify and solve $2x - 5 \ge 3(x - 4)$ .

Be careful with distribution and combining like terms

When you expand $3(2 - x)$ and $3(x - 4)$ , make sure you multiply 3 by both terms inside the parentheses, and then combine like terms correctly before solving.

Think about the overlap

Once you have each inequality solved in the form $x > \text{(number)}$ or $x \le \text{(number)}$ , think about the range of $x$ values that satisfy both conditions at the same time.

Focus on integers only at the end

After you know the full interval of $x$ values that work, list the integers in that interval and look for the largest one.

Desmos Guide

Graph the first inequality

Enter y = 3(2 - x) + 8 on one line and y = 5x - 4 on another. Look at the $x$ -values where the graph of $y = 3(2 - x) + 8$ is below the graph of $y = 5x - 4$ ; these $x$ -values satisfy the first inequality.

Graph the second inequality

Enter y = 2x - 5 on one line and y = 3(x - 4) on another. Look at the $x$ -values where the graph of y = 2x - 5 is on or above the graph of y = 3(x - 4); these $x$ -values satisfy the second inequality.

Find the overlap and check integers

Determine the range of $x$ where both conditions from Steps 1 and 2 are true (where the first line is below its partner and the second line is on/above its partner). Then use a table in Desmos or plug in integer $x$ -values in that overlapping region to see which integers work and identify the largest one.

Step-by-step Explanation

Solve the first inequality

Start with

3(2 - x) + 8 < 5x - 4.

Distribute the 3:

6 - 3x + 8 < 5x - 4.

Combine like terms on the left:

14 - 3x < 5x - 4.

Add $3x$ to both sides:

14 < 8x - 4.

Add 4 to both sides:

18 < 8x.

Divide both sides by 8:

\frac{18}{8} < x,

which simplifies to

\frac{9}{4} < x.

So the first inequality tells you $x$ must be greater than $9/4$ .

Solve the second inequality

Now solve

2x - 5 \ge 3(x - 4).

Distribute the 3 on the right:

2x - 5 \ge 3x - 12.

Subtract $2x$ from both sides:

-5 \ge x - 12.

Add 12 to both sides:

7 \ge x.

This means $x$ must be less than or equal to 7.

Combine the two conditions

From Step 1, you have $x > 9/4$ .

From Step 2, you have $x \le 7$ .

To satisfy both inequalities at the same time, $x$ must be in the intersection of these two sets:

Greater than $9/4$ (which is $2.25$ )
Less than or equal to 7

So $x$ must be between $9/4$ and 7, including 7 but not including $9/4$ .

The integers in this interval are $3, 4, 5, 6$ , and $7$ .

Choose the greatest integer that works

The question asks for the greatest integer value of $x$ that satisfies both inequalities.

From the previous step, the integers that work are $3, 4, 5, 6$ , and $7$ , and the greatest of these is 7.

So, the greatest integer value of $x$ that satisfies both inequalities is 7.

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Step-by-step Explanation

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