Question 630·Hard·Algebra

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For what value of $k$ is the inequality

(k-2)x + 7 \ge (3k + 1)x + 4

true for every real number $x$ ?

A linear inequality that must hold for every real $x$ needs a different test from one asking which values of $x$ work. If the two sides have different coefficients of $x$ , a large positive or negative $x$ will make it fail. Match the coefficients, check the numbers left over, and use a Desmos regression to find the parameter. Testing only $x=0$ misses the difference between the coefficients.

Hints

Hint 1
A coefficient is the number multiplying $x$ . If the coefficients on the two sides differ, a large positive or negative $x$ can make the inequality false. What must be true about the two coefficients?
Hint 2
After the $x$ -terms cancel, the inequality must still be true. Compare the numbers left over, then use a Desmos regression with $\sim$ in place of $=$ to find the coefficient-matching value of $k$ .

Step-by-step

Match the coefficients
Step 1
Make the $x$ -terms cancel
The coefficient of $x$ is the expression multiplying it: $k-2$ on the left and $3k+1$ on the right. If those coefficients differ, a sufficiently large positive or negative $x$ will make the inequality false. So every real $x$ requires
$k-2=3k+1.$
Step 2
Check what remains
When the coefficients match, the $x$ -terms cancel. The inequality becomes
$7\ge4.$
That's true, so matching the coefficients is enough here. If the numbers left over made a false inequality, no value of $k$ would work.
Step 3
Find the matching value of $k$
Type $k-2\sim3k+1$ . The $\sim$ tells Desmos to fit $k$ so the coefficients match. Under PARAMETERS, it shows $k=-1.5$ , or $-\frac{3}{2}$ . Since $7\ge4$ remains true, the inequality holds for every real $x$ when $k=-\frac{3}{2}$ . Choice D.

Question 630·Hard·Algebra

Strategy

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Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

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