SAT · Math · Algebra
Desigualdades lineales y sistemas de desigualdades, en abstracto o en contextos (presupuestos, límites de peso…).
Tienes 50 \$ y cada entrada cuesta 7 \$: $7n\le50$ → $n\le7{,}14$ → como máximo 7 entradas.
Intenta responder antes de abrir la solución. Las preguntas están en el idioma del examen.
Which of the following values of $x$ satisfies the inequality $5x-3>17$?
Respuesta correcta: D. $5$
Add 3 to both sides: $5x>20$. Divide by 5: $x>4$. Of the choices, only 5 is greater than 4.
4 makes the two sides equal ($5(4)-3=17$), and a strict inequality (>) does not include equality. 2 and 3 give 7 and 12, which are less than 17.
$$-2(x-4)\le 3x-7$$
What is the least possible value of $x$ that satisfies the given inequality?
Respuesta numérica (escribes el número).
Respuesta: 3
Distribute: $-2x+8\le 3x-7$. Add $2x$ and $7$ to both sides: $15\le 5x$. Divide by $5$: $3\le x$, so $x\ge 3$. The least possible value is $3$.
Check: at $x=3$, $-2(-1)=2$ and $3(3)-7=2$, so both sides are equal.
$$\begin{cases} y\le -2x+12 \\ y\ge x \end{cases}$$
The point $(a, b)$ lies in the solution region of the given system of inequalities in the $xy$-plane. What is the greatest possible value of $a$?
Respuesta numérica (escribes el número).
Respuesta: 4
For a point in the region, $x\le y\le -2x+12$, so $x\le -2x+12$. Add $2x$: $3x\le 12$, so $x\le 4$. The greatest possible value of $a$ is $4$, at the corner $(4, 4)$ where the two boundary lines meet.
Check: $(4, 4)$ satisfies $4\le -8+12=4$ and $4\ge 4$.